Linear Algebra Examples

$[\begin{array}{c} 7 & 2.12 & - 3.25 & 5.2 \\ 4 & 3 & 5 & - 2 \\ 1.23 & 5.21 & - 4.03 & 7.2 \\ 4.2 & 12.06 & - 3.085 & 4.74 \end{array}] \cdot 4 - 3.085$

Step 1

Simplify each term.

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Step 1.1

Move $4$ to the left of $[\begin{array}{c} 7 & 2.12 & - 3.25 & 5.2 \\ 4 & 3 & 5 & - 2 \\ 1.23 & 5.21 & - 4.03 & 7.2 \\ 4.2 & 12.06 & - 3.085 & 4.74 \end{array}]$ .

$4 \cdot [\begin{array}{c} 7 & 2.12 & - 3.25 & 5.2 \\ 4 & 3 & 5 & - 2 \\ 1.23 & 5.21 & - 4.03 & 7.2 \\ 4.2 & 12.06 & - 3.085 & 4.74 \end{array}] - 3.085$

Step 1.2

Multiply $4$ by each element of the matrix.

$[\begin{array}{c} 4 \cdot 7 & 4 \cdot 2.12 & 4 \cdot - 3.25 & 4 \cdot 5.2 \\ 4 \cdot 4 & 4 \cdot 3 & 4 \cdot 5 & 4 \cdot - 2 \\ 4 \cdot 1.23 & 4 \cdot 5.21 & 4 \cdot - 4.03 & 4 \cdot 7.2 \\ 4 \cdot 4.2 & 4 \cdot 12.06 & 4 \cdot - 3.085 & 4 \cdot 4.74 \end{array}] - 3.085$

Step 1.3

Simplify each element in the matrix.

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Step 1.3.1

Multiply $4$ by $7$ .

$[\begin{array}{c} 28 & 4 \cdot 2.12 & 4 \cdot - 3.25 & 4 \cdot 5.2 \\ 4 \cdot 4 & 4 \cdot 3 & 4 \cdot 5 & 4 \cdot - 2 \\ 4 \cdot 1.23 & 4 \cdot 5.21 & 4 \cdot - 4.03 & 4 \cdot 7.2 \\ 4 \cdot 4.2 & 4 \cdot 12.06 & 4 \cdot - 3.085 & 4 \cdot 4.74 \end{array}] - 3.085$

Step 1.3.2

Multiply $4$ by $2.12$ .

$[\begin{array}{c} 28 & 8.48 & 4 \cdot - 3.25 & 4 \cdot 5.2 \\ 4 \cdot 4 & 4 \cdot 3 & 4 \cdot 5 & 4 \cdot - 2 \\ 4 \cdot 1.23 & 4 \cdot 5.21 & 4 \cdot - 4.03 & 4 \cdot 7.2 \\ 4 \cdot 4.2 & 4 \cdot 12.06 & 4 \cdot - 3.085 & 4 \cdot 4.74 \end{array}] - 3.085$

Step 1.3.3

Multiply $4$ by $- 3.25$ .

$[\begin{array}{c} 28 & 8.48 & - 13 & 4 \cdot 5.2 \\ 4 \cdot 4 & 4 \cdot 3 & 4 \cdot 5 & 4 \cdot - 2 \\ 4 \cdot 1.23 & 4 \cdot 5.21 & 4 \cdot - 4.03 & 4 \cdot 7.2 \\ 4 \cdot 4.2 & 4 \cdot 12.06 & 4 \cdot - 3.085 & 4 \cdot 4.74 \end{array}] - 3.085$

Step 1.3.4

Multiply $4$ by $5.2$ .

$[\begin{array}{c} 28 & 8.48 & - 13 & 20.8 \\ 4 \cdot 4 & 4 \cdot 3 & 4 \cdot 5 & 4 \cdot - 2 \\ 4 \cdot 1.23 & 4 \cdot 5.21 & 4 \cdot - 4.03 & 4 \cdot 7.2 \\ 4 \cdot 4.2 & 4 \cdot 12.06 & 4 \cdot - 3.085 & 4 \cdot 4.74 \end{array}] - 3.085$

Step 1.3.5

Multiply $4$ by $4$ .

$[\begin{array}{c} 28 & 8.48 & - 13 & 20.8 \\ 16 & 4 \cdot 3 & 4 \cdot 5 & 4 \cdot - 2 \\ 4 \cdot 1.23 & 4 \cdot 5.21 & 4 \cdot - 4.03 & 4 \cdot 7.2 \\ 4 \cdot 4.2 & 4 \cdot 12.06 & 4 \cdot - 3.085 & 4 \cdot 4.74 \end{array}] - 3.085$

Step 1.3.6

Multiply $4$ by $3$ .

$[\begin{array}{c} 28 & 8.48 & - 13 & 20.8 \\ 16 & 12 & 4 \cdot 5 & 4 \cdot - 2 \\ 4 \cdot 1.23 & 4 \cdot 5.21 & 4 \cdot - 4.03 & 4 \cdot 7.2 \\ 4 \cdot 4.2 & 4 \cdot 12.06 & 4 \cdot - 3.085 & 4 \cdot 4.74 \end{array}] - 3.085$

Step 1.3.7

Multiply $4$ by $5$ .

$[\begin{array}{c} 28 & 8.48 & - 13 & 20.8 \\ 16 & 12 & 20 & 4 \cdot - 2 \\ 4 \cdot 1.23 & 4 \cdot 5.21 & 4 \cdot - 4.03 & 4 \cdot 7.2 \\ 4 \cdot 4.2 & 4 \cdot 12.06 & 4 \cdot - 3.085 & 4 \cdot 4.74 \end{array}] - 3.085$

Step 1.3.8

Multiply $4$ by $- 2$ .

$[\begin{array}{c} 28 & 8.48 & - 13 & 20.8 \\ 16 & 12 & 20 & - 8 \\ 4 \cdot 1.23 & 4 \cdot 5.21 & 4 \cdot - 4.03 & 4 \cdot 7.2 \\ 4 \cdot 4.2 & 4 \cdot 12.06 & 4 \cdot - 3.085 & 4 \cdot 4.74 \end{array}] - 3.085$

Step 1.3.9

Multiply $4$ by $1.23$ .

$[\begin{array}{c} 28 & 8.48 & - 13 & 20.8 \\ 16 & 12 & 20 & - 8 \\ 4.92 & 4 \cdot 5.21 & 4 \cdot - 4.03 & 4 \cdot 7.2 \\ 4 \cdot 4.2 & 4 \cdot 12.06 & 4 \cdot - 3.085 & 4 \cdot 4.74 \end{array}] - 3.085$

Step 1.3.10

Multiply $4$ by $5.21$ .

$[\begin{array}{c} 28 & 8.48 & - 13 & 20.8 \\ 16 & 12 & 20 & - 8 \\ 4.92 & 20.84 & 4 \cdot - 4.03 & 4 \cdot 7.2 \\ 4 \cdot 4.2 & 4 \cdot 12.06 & 4 \cdot - 3.085 & 4 \cdot 4.74 \end{array}] - 3.085$

Step 1.3.11

Multiply $4$ by $- 4.03$ .

$[\begin{array}{c} 28 & 8.48 & - 13 & 20.8 \\ 16 & 12 & 20 & - 8 \\ 4.92 & 20.84 & - 16.12 & 4 \cdot 7.2 \\ 4 \cdot 4.2 & 4 \cdot 12.06 & 4 \cdot - 3.085 & 4 \cdot 4.74 \end{array}] - 3.085$

Step 1.3.12

Multiply $4$ by $7.2$ .

$[\begin{array}{c} 28 & 8.48 & - 13 & 20.8 \\ 16 & 12 & 20 & - 8 \\ 4.92 & 20.84 & - 16.12 & 28.8 \\ 4 \cdot 4.2 & 4 \cdot 12.06 & 4 \cdot - 3.085 & 4 \cdot 4.74 \end{array}] - 3.085$

Step 1.3.13

Multiply $4$ by $4.2$ .

$[\begin{array}{c} 28 & 8.48 & - 13 & 20.8 \\ 16 & 12 & 20 & - 8 \\ 4.92 & 20.84 & - 16.12 & 28.8 \\ 16.8 & 4 \cdot 12.06 & 4 \cdot - 3.085 & 4 \cdot 4.74 \end{array}] - 3.085$

Step 1.3.14

Multiply $4$ by $12.06$ .

$[\begin{array}{c} 28 & 8.48 & - 13 & 20.8 \\ 16 & 12 & 20 & - 8 \\ 4.92 & 20.84 & - 16.12 & 28.8 \\ 16.8 & 48.24 & 4 \cdot - 3.085 & 4 \cdot 4.74 \end{array}] - 3.085$

Step 1.3.15

Multiply $4$ by $- 3.085$ .

$[\begin{array}{c} 28 & 8.48 & - 13 & 20.8 \\ 16 & 12 & 20 & - 8 \\ 4.92 & 20.84 & - 16.12 & 28.8 \\ 16.8 & 48.24 & - 12.34 & 4 \cdot 4.74 \end{array}] - 3.085$

Step 1.3.16

Multiply $4$ by $4.74$ .

$[\begin{array}{c} 28 & 8.48 & - 13 & 20.8 \\ 16 & 12 & 20 & - 8 \\ 4.92 & 20.84 & - 16.12 & 28.8 \\ 16.8 & 48.24 & - 12.34 & 18.96 \end{array}] - 3.085$

Step 2

Adding $- 3.085$ to a $4 \times 4$ square matrix is the same as adding $- 3.085$ times the $4 \times 4$ identity matrix.

$[\begin{array}{c} 28 & 8.48 & - 13 & 20.8 \\ 16 & 12 & 20 & - 8 \\ 4.92 & 20.84 & - 16.12 & 28.8 \\ 16.8 & 48.24 & - 12.34 & 18.96 \end{array}] - 3.085 [\begin{array}{c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}]$

Step 3

Multiply $- 3.085$ by each element of the matrix.

$[\begin{array}{c} 28 & 8.48 & - 13 & 20.8 \\ 16 & 12 & 20 & - 8 \\ 4.92 & 20.84 & - 16.12 & 28.8 \\ 16.8 & 48.24 & - 12.34 & 18.96 \end{array}] + [\begin{array}{c} - 3.085 \cdot 1 & - 3.085 \cdot 0 & - 3.085 \cdot 0 & - 3.085 \cdot 0 \\ - 3.085 \cdot 0 & - 3.085 \cdot 1 & - 3.085 \cdot 0 & - 3.085 \cdot 0 \\ - 3.085 \cdot 0 & - 3.085 \cdot 0 & - 3.085 \cdot 1 & - 3.085 \cdot 0 \\ - 3.085 \cdot 0 & - 3.085 \cdot 0 & - 3.085 \cdot 0 & - 3.085 \cdot 1 \end{array}]$

Step 4

Simplify each element in the matrix.

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Step 4.1

Multiply $- 3.085$ by $1$ .

$[\begin{array}{c} 28 & 8.48 & - 13 & 20.8 \\ 16 & 12 & 20 & - 8 \\ 4.92 & 20.84 & - 16.12 & 28.8 \\ 16.8 & 48.24 & - 12.34 & 18.96 \end{array}] + [\begin{array}{c} - 3.085 & - 3.085 \cdot 0 & - 3.085 \cdot 0 & - 3.085 \cdot 0 \\ - 3.085 \cdot 0 & - 3.085 \cdot 1 & - 3.085 \cdot 0 & - 3.085 \cdot 0 \\ - 3.085 \cdot 0 & - 3.085 \cdot 0 & - 3.085 \cdot 1 & - 3.085 \cdot 0 \\ - 3.085 \cdot 0 & - 3.085 \cdot 0 & - 3.085 \cdot 0 & - 3.085 \cdot 1 \end{array}]$

Step 4.2

Multiply $- 3.085$ by $0$ .

$[\begin{array}{c} 28 & 8.48 & - 13 & 20.8 \\ 16 & 12 & 20 & - 8 \\ 4.92 & 20.84 & - 16.12 & 28.8 \\ 16.8 & 48.24 & - 12.34 & 18.96 \end{array}] + [\begin{array}{c} - 3.085 & 0 & - 3.085 \cdot 0 & - 3.085 \cdot 0 \\ - 3.085 \cdot 0 & - 3.085 \cdot 1 & - 3.085 \cdot 0 & - 3.085 \cdot 0 \\ - 3.085 \cdot 0 & - 3.085 \cdot 0 & - 3.085 \cdot 1 & - 3.085 \cdot 0 \\ - 3.085 \cdot 0 & - 3.085 \cdot 0 & - 3.085 \cdot 0 & - 3.085 \cdot 1 \end{array}]$

Step 4.3

Multiply $- 3.085$ by $0$ .

$[\begin{array}{c} 28 & 8.48 & - 13 & 20.8 \\ 16 & 12 & 20 & - 8 \\ 4.92 & 20.84 & - 16.12 & 28.8 \\ 16.8 & 48.24 & - 12.34 & 18.96 \end{array}] + [\begin{array}{c} - 3.085 & 0 & 0 & - 3.085 \cdot 0 \\ - 3.085 \cdot 0 & - 3.085 \cdot 1 & - 3.085 \cdot 0 & - 3.085 \cdot 0 \\ - 3.085 \cdot 0 & - 3.085 \cdot 0 & - 3.085 \cdot 1 & - 3.085 \cdot 0 \\ - 3.085 \cdot 0 & - 3.085 \cdot 0 & - 3.085 \cdot 0 & - 3.085 \cdot 1 \end{array}]$

Step 4.4

Multiply $- 3.085$ by $0$ .

$[\begin{array}{c} 28 & 8.48 & - 13 & 20.8 \\ 16 & 12 & 20 & - 8 \\ 4.92 & 20.84 & - 16.12 & 28.8 \\ 16.8 & 48.24 & - 12.34 & 18.96 \end{array}] + [\begin{array}{c} - 3.085 & 0 & 0 & 0 \\ - 3.085 \cdot 0 & - 3.085 \cdot 1 & - 3.085 \cdot 0 & - 3.085 \cdot 0 \\ - 3.085 \cdot 0 & - 3.085 \cdot 0 & - 3.085 \cdot 1 & - 3.085 \cdot 0 \\ - 3.085 \cdot 0 & - 3.085 \cdot 0 & - 3.085 \cdot 0 & - 3.085 \cdot 1 \end{array}]$

Step 4.5

Multiply $- 3.085$ by $0$ .

$[\begin{array}{c} 28 & 8.48 & - 13 & 20.8 \\ 16 & 12 & 20 & - 8 \\ 4.92 & 20.84 & - 16.12 & 28.8 \\ 16.8 & 48.24 & - 12.34 & 18.96 \end{array}] + [\begin{array}{c} - 3.085 & 0 & 0 & 0 \\ 0 & - 3.085 \cdot 1 & - 3.085 \cdot 0 & - 3.085 \cdot 0 \\ - 3.085 \cdot 0 & - 3.085 \cdot 0 & - 3.085 \cdot 1 & - 3.085 \cdot 0 \\ - 3.085 \cdot 0 & - 3.085 \cdot 0 & - 3.085 \cdot 0 & - 3.085 \cdot 1 \end{array}]$

Step 4.6

Multiply $- 3.085$ by $1$ .

$[\begin{array}{c} 28 & 8.48 & - 13 & 20.8 \\ 16 & 12 & 20 & - 8 \\ 4.92 & 20.84 & - 16.12 & 28.8 \\ 16.8 & 48.24 & - 12.34 & 18.96 \end{array}] + [\begin{array}{c} - 3.085 & 0 & 0 & 0 \\ 0 & - 3.085 & - 3.085 \cdot 0 & - 3.085 \cdot 0 \\ - 3.085 \cdot 0 & - 3.085 \cdot 0 & - 3.085 \cdot 1 & - 3.085 \cdot 0 \\ - 3.085 \cdot 0 & - 3.085 \cdot 0 & - 3.085 \cdot 0 & - 3.085 \cdot 1 \end{array}]$

Step 4.7

Multiply $- 3.085$ by $0$ .

$[\begin{array}{c} 28 & 8.48 & - 13 & 20.8 \\ 16 & 12 & 20 & - 8 \\ 4.92 & 20.84 & - 16.12 & 28.8 \\ 16.8 & 48.24 & - 12.34 & 18.96 \end{array}] + [\begin{array}{c} - 3.085 & 0 & 0 & 0 \\ 0 & - 3.085 & 0 & - 3.085 \cdot 0 \\ - 3.085 \cdot 0 & - 3.085 \cdot 0 & - 3.085 \cdot 1 & - 3.085 \cdot 0 \\ - 3.085 \cdot 0 & - 3.085 \cdot 0 & - 3.085 \cdot 0 & - 3.085 \cdot 1 \end{array}]$

Step 4.8

Multiply $- 3.085$ by $0$ .

$[\begin{array}{c} 28 & 8.48 & - 13 & 20.8 \\ 16 & 12 & 20 & - 8 \\ 4.92 & 20.84 & - 16.12 & 28.8 \\ 16.8 & 48.24 & - 12.34 & 18.96 \end{array}] + [\begin{array}{c} - 3.085 & 0 & 0 & 0 \\ 0 & - 3.085 & 0 & 0 \\ - 3.085 \cdot 0 & - 3.085 \cdot 0 & - 3.085 \cdot 1 & - 3.085 \cdot 0 \\ - 3.085 \cdot 0 & - 3.085 \cdot 0 & - 3.085 \cdot 0 & - 3.085 \cdot 1 \end{array}]$

Step 4.9

Multiply $- 3.085$ by $0$ .

$[\begin{array}{c} 28 & 8.48 & - 13 & 20.8 \\ 16 & 12 & 20 & - 8 \\ 4.92 & 20.84 & - 16.12 & 28.8 \\ 16.8 & 48.24 & - 12.34 & 18.96 \end{array}] + [\begin{array}{c} - 3.085 & 0 & 0 & 0 \\ 0 & - 3.085 & 0 & 0 \\ 0 & - 3.085 \cdot 0 & - 3.085 \cdot 1 & - 3.085 \cdot 0 \\ - 3.085 \cdot 0 & - 3.085 \cdot 0 & - 3.085 \cdot 0 & - 3.085 \cdot 1 \end{array}]$

Step 4.10

Multiply $- 3.085$ by $0$ .

$[\begin{array}{c} 28 & 8.48 & - 13 & 20.8 \\ 16 & 12 & 20 & - 8 \\ 4.92 & 20.84 & - 16.12 & 28.8 \\ 16.8 & 48.24 & - 12.34 & 18.96 \end{array}] + [\begin{array}{c} - 3.085 & 0 & 0 & 0 \\ 0 & - 3.085 & 0 & 0 \\ 0 & 0 & - 3.085 \cdot 1 & - 3.085 \cdot 0 \\ - 3.085 \cdot 0 & - 3.085 \cdot 0 & - 3.085 \cdot 0 & - 3.085 \cdot 1 \end{array}]$

Step 4.11

Multiply $- 3.085$ by $1$ .

$[\begin{array}{c} 28 & 8.48 & - 13 & 20.8 \\ 16 & 12 & 20 & - 8 \\ 4.92 & 20.84 & - 16.12 & 28.8 \\ 16.8 & 48.24 & - 12.34 & 18.96 \end{array}] + [\begin{array}{c} - 3.085 & 0 & 0 & 0 \\ 0 & - 3.085 & 0 & 0 \\ 0 & 0 & - 3.085 & - 3.085 \cdot 0 \\ - 3.085 \cdot 0 & - 3.085 \cdot 0 & - 3.085 \cdot 0 & - 3.085 \cdot 1 \end{array}]$

Step 4.12

Multiply $- 3.085$ by $0$ .

$[\begin{array}{c} 28 & 8.48 & - 13 & 20.8 \\ 16 & 12 & 20 & - 8 \\ 4.92 & 20.84 & - 16.12 & 28.8 \\ 16.8 & 48.24 & - 12.34 & 18.96 \end{array}] + [\begin{array}{c} - 3.085 & 0 & 0 & 0 \\ 0 & - 3.085 & 0 & 0 \\ 0 & 0 & - 3.085 & 0 \\ - 3.085 \cdot 0 & - 3.085 \cdot 0 & - 3.085 \cdot 0 & - 3.085 \cdot 1 \end{array}]$

Step 4.13

Multiply $- 3.085$ by $0$ .

$[\begin{array}{c} 28 & 8.48 & - 13 & 20.8 \\ 16 & 12 & 20 & - 8 \\ 4.92 & 20.84 & - 16.12 & 28.8 \\ 16.8 & 48.24 & - 12.34 & 18.96 \end{array}] + [\begin{array}{c} - 3.085 & 0 & 0 & 0 \\ 0 & - 3.085 & 0 & 0 \\ 0 & 0 & - 3.085 & 0 \\ 0 & - 3.085 \cdot 0 & - 3.085 \cdot 0 & - 3.085 \cdot 1 \end{array}]$

Step 4.14

Multiply $- 3.085$ by $0$ .

$[\begin{array}{c} 28 & 8.48 & - 13 & 20.8 \\ 16 & 12 & 20 & - 8 \\ 4.92 & 20.84 & - 16.12 & 28.8 \\ 16.8 & 48.24 & - 12.34 & 18.96 \end{array}] + [\begin{array}{c} - 3.085 & 0 & 0 & 0 \\ 0 & - 3.085 & 0 & 0 \\ 0 & 0 & - 3.085 & 0 \\ 0 & 0 & - 3.085 \cdot 0 & - 3.085 \cdot 1 \end{array}]$

Step 4.15

Multiply $- 3.085$ by $0$ .

$[\begin{array}{c} 28 & 8.48 & - 13 & 20.8 \\ 16 & 12 & 20 & - 8 \\ 4.92 & 20.84 & - 16.12 & 28.8 \\ 16.8 & 48.24 & - 12.34 & 18.96 \end{array}] + [\begin{array}{c} - 3.085 & 0 & 0 & 0 \\ 0 & - 3.085 & 0 & 0 \\ 0 & 0 & - 3.085 & 0 \\ 0 & 0 & 0 & - 3.085 \cdot 1 \end{array}]$

Step 4.16

Multiply $- 3.085$ by $1$ .

$[\begin{array}{c} 28 & 8.48 & - 13 & 20.8 \\ 16 & 12 & 20 & - 8 \\ 4.92 & 20.84 & - 16.12 & 28.8 \\ 16.8 & 48.24 & - 12.34 & 18.96 \end{array}] + [\begin{array}{c} - 3.085 & 0 & 0 & 0 \\ 0 & - 3.085 & 0 & 0 \\ 0 & 0 & - 3.085 & 0 \\ 0 & 0 & 0 & - 3.085 \end{array}]$

Step 5

Add the corresponding elements.

$[\begin{array}{c} 28 - 3.085 & 8.48 + 0 & - 13 + 0 & 20.8 + 0 \\ 16 + 0 & 12 - 3.085 & 20 + 0 & - 8 + 0 \\ 4.92 + 0 & 20.84 + 0 & - 16.12 - 3.085 & 28.8 + 0 \\ 16.8 + 0 & 48.24 + 0 & - 12.34 + 0 & 18.96 - 3.085 \end{array}]$

Step 6

Simplify each element.

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Step 6.1

Subtract $3.085$ from $28$ .

$[\begin{array}{c} 24.915 & 8.48 + 0 & - 13 + 0 & 20.8 + 0 \\ 16 + 0 & 12 - 3.085 & 20 + 0 & - 8 + 0 \\ 4.92 + 0 & 20.84 + 0 & - 16.12 - 3.085 & 28.8 + 0 \\ 16.8 + 0 & 48.24 + 0 & - 12.34 + 0 & 18.96 - 3.085 \end{array}]$

Step 6.2

Add $8.48$ and $0$ .

$[\begin{array}{c} 24.915 & 8.48 & - 13 + 0 & 20.8 + 0 \\ 16 + 0 & 12 - 3.085 & 20 + 0 & - 8 + 0 \\ 4.92 + 0 & 20.84 + 0 & - 16.12 - 3.085 & 28.8 + 0 \\ 16.8 + 0 & 48.24 + 0 & - 12.34 + 0 & 18.96 - 3.085 \end{array}]$

Step 6.3

Add $- 13$ and $0$ .

$[\begin{array}{c} 24.915 & 8.48 & - 13 & 20.8 + 0 \\ 16 + 0 & 12 - 3.085 & 20 + 0 & - 8 + 0 \\ 4.92 + 0 & 20.84 + 0 & - 16.12 - 3.085 & 28.8 + 0 \\ 16.8 + 0 & 48.24 + 0 & - 12.34 + 0 & 18.96 - 3.085 \end{array}]$

Step 6.4

Add $20.8$ and $0$ .

$[\begin{array}{c} 24.915 & 8.48 & - 13 & 20.8 \\ 16 + 0 & 12 - 3.085 & 20 + 0 & - 8 + 0 \\ 4.92 + 0 & 20.84 + 0 & - 16.12 - 3.085 & 28.8 + 0 \\ 16.8 + 0 & 48.24 + 0 & - 12.34 + 0 & 18.96 - 3.085 \end{array}]$

Step 6.5

Add $16$ and $0$ .

$[\begin{array}{c} 24.915 & 8.48 & - 13 & 20.8 \\ 16 & 12 - 3.085 & 20 + 0 & - 8 + 0 \\ 4.92 + 0 & 20.84 + 0 & - 16.12 - 3.085 & 28.8 + 0 \\ 16.8 + 0 & 48.24 + 0 & - 12.34 + 0 & 18.96 - 3.085 \end{array}]$

Step 6.6

Subtract $3.085$ from $12$ .

$[\begin{array}{c} 24.915 & 8.48 & - 13 & 20.8 \\ 16 & 8.915 & 20 + 0 & - 8 + 0 \\ 4.92 + 0 & 20.84 + 0 & - 16.12 - 3.085 & 28.8 + 0 \\ 16.8 + 0 & 48.24 + 0 & - 12.34 + 0 & 18.96 - 3.085 \end{array}]$

Step 6.7

Add $20$ and $0$ .

$[\begin{array}{c} 24.915 & 8.48 & - 13 & 20.8 \\ 16 & 8.915 & 20 & - 8 + 0 \\ 4.92 + 0 & 20.84 + 0 & - 16.12 - 3.085 & 28.8 + 0 \\ 16.8 + 0 & 48.24 + 0 & - 12.34 + 0 & 18.96 - 3.085 \end{array}]$

Step 6.8

Add $- 8$ and $0$ .

$[\begin{array}{c} 24.915 & 8.48 & - 13 & 20.8 \\ 16 & 8.915 & 20 & - 8 \\ 4.92 + 0 & 20.84 + 0 & - 16.12 - 3.085 & 28.8 + 0 \\ 16.8 + 0 & 48.24 + 0 & - 12.34 + 0 & 18.96 - 3.085 \end{array}]$

Step 6.9

Add $4.92$ and $0$ .

$[\begin{array}{c} 24.915 & 8.48 & - 13 & 20.8 \\ 16 & 8.915 & 20 & - 8 \\ 4.92 & 20.84 + 0 & - 16.12 - 3.085 & 28.8 + 0 \\ 16.8 + 0 & 48.24 + 0 & - 12.34 + 0 & 18.96 - 3.085 \end{array}]$

Step 6.10

Add $20.84$ and $0$ .

$[\begin{array}{c} 24.915 & 8.48 & - 13 & 20.8 \\ 16 & 8.915 & 20 & - 8 \\ 4.92 & 20.84 & - 16.12 - 3.085 & 28.8 + 0 \\ 16.8 + 0 & 48.24 + 0 & - 12.34 + 0 & 18.96 - 3.085 \end{array}]$

Step 6.11

Subtract $3.085$ from $- 16.12$ .

$[\begin{array}{c} 24.915 & 8.48 & - 13 & 20.8 \\ 16 & 8.915 & 20 & - 8 \\ 4.92 & 20.84 & - 19.205 & 28.8 + 0 \\ 16.8 + 0 & 48.24 + 0 & - 12.34 + 0 & 18.96 - 3.085 \end{array}]$

Step 6.12

Add $28.8$ and $0$ .

$[\begin{array}{c} 24.915 & 8.48 & - 13 & 20.8 \\ 16 & 8.915 & 20 & - 8 \\ 4.92 & 20.84 & - 19.205 & 28.8 \\ 16.8 + 0 & 48.24 + 0 & - 12.34 + 0 & 18.96 - 3.085 \end{array}]$

Step 6.13

Add $16.8$ and $0$ .

$[\begin{array}{c} 24.915 & 8.48 & - 13 & 20.8 \\ 16 & 8.915 & 20 & - 8 \\ 4.92 & 20.84 & - 19.205 & 28.8 \\ 16.8 & 48.24 + 0 & - 12.34 + 0 & 18.96 - 3.085 \end{array}]$

Step 6.14

Add $48.24$ and $0$ .

$[\begin{array}{c} 24.915 & 8.48 & - 13 & 20.8 \\ 16 & 8.915 & 20 & - 8 \\ 4.92 & 20.84 & - 19.205 & 28.8 \\ 16.8 & 48.24 & - 12.34 + 0 & 18.96 - 3.085 \end{array}]$

Step 6.15

Add $- 12.34$ and $0$ .

$[\begin{array}{c} 24.915 & 8.48 & - 13 & 20.8 \\ 16 & 8.915 & 20 & - 8 \\ 4.92 & 20.84 & - 19.205 & 28.8 \\ 16.8 & 48.24 & - 12.34 & 18.96 - 3.085 \end{array}]$

Step 6.16

Subtract $3.085$ from $18.96$ .

$[\begin{array}{c} 24.915 & 8.48 & - 13 & 20.8 \\ 16 & 8.915 & 20 & - 8 \\ 4.92 & 20.84 & - 19.205 & 28.8 \\ 16.8 & 48.24 & - 12.34 & 15.875 \end{array}]$

Step 7

Choose the row or column with the most $0$ elements. If there are no $0$ elements choose any row or column. Multiply every element in row $1$ by its cofactor and add.

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Step 7.1

Consider the corresponding sign chart.

$| \begin{array}{c} + & - & + & - \\ - & + & - & + \\ + & - & + & - \\ - & + & - & + \end{array} |$

Step 7.2

The cofactor is the minor with the sign changed if the indices match a $-$ position on the sign chart.

Step 7.3

The minor for $a_{11}$ is the determinant with row $1$ and column $1$ deleted.

$| \begin{array}{c} 8.915 & 20 & - 8 \\ 20.84 & - 19.205 & 28.8 \\ 48.24 & - 12.34 & 15.875 \end{array} |$

Step 7.4

Multiply element $a_{11}$ by its cofactor.

$24.915 | \begin{array}{c} 8.915 & 20 & - 8 \\ 20.84 & - 19.205 & 28.8 \\ 48.24 & - 12.34 & 15.875 \end{array} |$

Step 7.5

The minor for $a_{12}$ is the determinant with row $1$ and column $2$ deleted.

$| \begin{array}{c} 16 & 20 & - 8 \\ 4.92 & - 19.205 & 28.8 \\ 16.8 & - 12.34 & 15.875 \end{array} |$

Step 7.6

Multiply element $a_{12}$ by its cofactor.

$- 8.48 | \begin{array}{c} 16 & 20 & - 8 \\ 4.92 & - 19.205 & 28.8 \\ 16.8 & - 12.34 & 15.875 \end{array} |$

Step 7.7

The minor for $a_{13}$ is the determinant with row $1$ and column $3$ deleted.

$| \begin{array}{c} 16 & 8.915 & - 8 \\ 4.92 & 20.84 & 28.8 \\ 16.8 & 48.24 & 15.875 \end{array} |$

Step 7.8

Multiply element $a_{13}$ by its cofactor.

$- 13 | \begin{array}{c} 16 & 8.915 & - 8 \\ 4.92 & 20.84 & 28.8 \\ 16.8 & 48.24 & 15.875 \end{array} |$

Step 7.9

The minor for $a_{14}$ is the determinant with row $1$ and column $4$ deleted.

$| \begin{array}{c} 16 & 8.915 & 20 \\ 4.92 & 20.84 & - 19.205 \\ 16.8 & 48.24 & - 12.34 \end{array} |$

Step 7.10

Multiply element $a_{14}$ by its cofactor.

$- 20.8 | \begin{array}{c} 16 & 8.915 & 20 \\ 4.92 & 20.84 & - 19.205 \\ 16.8 & 48.24 & - 12.34 \end{array} |$

Step 7.11

Add the terms together.

$24.915 | \begin{array}{c} 8.915 & 20 & - 8 \\ 20.84 & - 19.205 & 28.8 \\ 48.24 & - 12.34 & 15.875 \end{array} | - 8.48 | \begin{array}{c} 16 & 20 & - 8 \\ 4.92 & - 19.205 & 28.8 \\ 16.8 & - 12.34 & 15.875 \end{array} | - 13 | \begin{array}{c} 16 & 8.915 & - 8 \\ 4.92 & 20.84 & 28.8 \\ 16.8 & 48.24 & 15.875 \end{array} | - 20.8 | \begin{array}{c} 16 & 8.915 & 20 \\ 4.92 & 20.84 & - 19.205 \\ 16.8 & 48.24 & - 12.34 \end{array} |$

Step 8

Evaluate $| \begin{array}{c} 8.915 & 20 & - 8 \\ 20.84 & - 19.205 & 28.8 \\ 48.24 & - 12.34 & 15.875 \end{array} |$ .

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Step 8.1

Choose the row or column with the most $0$ elements. If there are no $0$ elements choose any row or column. Multiply every element in row $1$ by its cofactor and add.

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Step 8.1.1

Consider the corresponding sign chart.

$| \begin{array}{c} + & - & + \\ - & + & - \\ + & - & + \end{array} |$

Step 8.1.2

The cofactor is the minor with the sign changed if the indices match a $-$ position on the sign chart.

Step 8.1.3

The minor for $a_{11}$ is the determinant with row $1$ and column $1$ deleted.

$| \begin{array}{c} - 19.205 & 28.8 \\ - 12.34 & 15.875 \end{array} |$

Step 8.1.4

Multiply element $a_{11}$ by its cofactor.

$8.915 | \begin{array}{c} - 19.205 & 28.8 \\ - 12.34 & 15.875 \end{array} |$

Step 8.1.5

The minor for $a_{12}$ is the determinant with row $1$ and column $2$ deleted.

$| \begin{array}{c} 20.84 & 28.8 \\ 48.24 & 15.875 \end{array} |$

Step 8.1.6

Multiply element $a_{12}$ by its cofactor.

$- 20 | \begin{array}{c} 20.84 & 28.8 \\ 48.24 & 15.875 \end{array} |$

Step 8.1.7

The minor for $a_{13}$ is the determinant with row $1$ and column $3$ deleted.

$| \begin{array}{c} 20.84 & - 19.205 \\ 48.24 & - 12.34 \end{array} |$

Step 8.1.8

Multiply element $a_{13}$ by its cofactor.

$- 8 | \begin{array}{c} 20.84 & - 19.205 \\ 48.24 & - 12.34 \end{array} |$

Step 8.1.9

Add the terms together.

$24.915 (8.915 | \begin{array}{c} - 19.205 & 28.8 \\ - 12.34 & 15.875 \end{array} | - 20 | \begin{array}{c} 20.84 & 28.8 \\ 48.24 & 15.875 \end{array} | - 8 | \begin{array}{c} 20.84 & - 19.205 \\ 48.24 & - 12.34 \end{array} |) - 8.48 | \begin{array}{c} 16 & 20 & - 8 \\ 4.92 & - 19.205 & 28.8 \\ 16.8 & - 12.34 & 15.875 \end{array} | - 13 | \begin{array}{c} 16 & 8.915 & - 8 \\ 4.92 & 20.84 & 28.8 \\ 16.8 & 48.24 & 15.875 \end{array} | - 20.8 | \begin{array}{c} 16 & 8.915 & 20 \\ 4.92 & 20.84 & - 19.205 \\ 16.8 & 48.24 & - 12.34 \end{array} |$

Step 8.2

Evaluate $| \begin{array}{c} - 19.205 & 28.8 \\ - 12.34 & 15.875 \end{array} |$ .

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Step 8.2.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$24.915 (8.915 (- 19.205 \cdot 15.875 - (- 12.34 \cdot 28.8)) - 20 | \begin{array}{c} 20.84 & 28.8 \\ 48.24 & 15.875 \end{array} | - 8 | \begin{array}{c} 20.84 & - 19.205 \\ 48.24 & - 12.34 \end{array} |) - 8.48 | \begin{array}{c} 16 & 20 & - 8 \\ 4.92 & - 19.205 & 28.8 \\ 16.8 & - 12.34 & 15.875 \end{array} | - 13 | \begin{array}{c} 16 & 8.915 & - 8 \\ 4.92 & 20.84 & 28.8 \\ 16.8 & 48.24 & 15.875 \end{array} | - 20.8 | \begin{array}{c} 16 & 8.915 & 20 \\ 4.92 & 20.84 & - 19.205 \\ 16.8 & 48.24 & - 12.34 \end{array} |$

Step 8.2.2

Simplify the determinant.

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Step 8.2.2.1

Simplify each term.

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Step 8.2.2.1.1

Multiply $- 19.205$ by $15.875$ .

$24.915 (8.915 (- 304.879375 - (- 12.34 \cdot 28.8)) - 20 | \begin{array}{c} 20.84 & 28.8 \\ 48.24 & 15.875 \end{array} | - 8 | \begin{array}{c} 20.84 & - 19.205 \\ 48.24 & - 12.34 \end{array} |) - 8.48 | \begin{array}{c} 16 & 20 & - 8 \\ 4.92 & - 19.205 & 28.8 \\ 16.8 & - 12.34 & 15.875 \end{array} | - 13 | \begin{array}{c} 16 & 8.915 & - 8 \\ 4.92 & 20.84 & 28.8 \\ 16.8 & 48.24 & 15.875 \end{array} | - 20.8 | \begin{array}{c} 16 & 8.915 & 20 \\ 4.92 & 20.84 & - 19.205 \\ 16.8 & 48.24 & - 12.34 \end{array} |$

Step 8.2.2.1.2

Multiply $- (- 12.34 \cdot 28.8)$ .

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Step 8.2.2.1.2.1

Multiply $- 12.34$ by $28.8$ .

$24.915 (8.915 (- 304.879375 - - 355.392) - 20 | \begin{array}{c} 20.84 & 28.8 \\ 48.24 & 15.875 \end{array} | - 8 | \begin{array}{c} 20.84 & - 19.205 \\ 48.24 & - 12.34 \end{array} |) - 8.48 | \begin{array}{c} 16 & 20 & - 8 \\ 4.92 & - 19.205 & 28.8 \\ 16.8 & - 12.34 & 15.875 \end{array} | - 13 | \begin{array}{c} 16 & 8.915 & - 8 \\ 4.92 & 20.84 & 28.8 \\ 16.8 & 48.24 & 15.875 \end{array} | - 20.8 | \begin{array}{c} 16 & 8.915 & 20 \\ 4.92 & 20.84 & - 19.205 \\ 16.8 & 48.24 & - 12.34 \end{array} |$

Step 8.2.2.1.2.2

Multiply $- 1$ by $- 355.392$ .

$24.915 (8.915 (- 304.879375 + 355.392) - 20 | \begin{array}{c} 20.84 & 28.8 \\ 48.24 & 15.875 \end{array} | - 8 | \begin{array}{c} 20.84 & - 19.205 \\ 48.24 & - 12.34 \end{array} |) - 8.48 | \begin{array}{c} 16 & 20 & - 8 \\ 4.92 & - 19.205 & 28.8 \\ 16.8 & - 12.34 & 15.875 \end{array} | - 13 | \begin{array}{c} 16 & 8.915 & - 8 \\ 4.92 & 20.84 & 28.8 \\ 16.8 & 48.24 & 15.875 \end{array} | - 20.8 | \begin{array}{c} 16 & 8.915 & 20 \\ 4.92 & 20.84 & - 19.205 \\ 16.8 & 48.24 & - 12.34 \end{array} |$

Step 8.2.2.2

Add $- 304.879375$ and $355.392$ .

$24.915 (8.915 \cdot 50.512625 - 20 | \begin{array}{c} 20.84 & 28.8 \\ 48.24 & 15.875 \end{array} | - 8 | \begin{array}{c} 20.84 & - 19.205 \\ 48.24 & - 12.34 \end{array} |) - 8.48 | \begin{array}{c} 16 & 20 & - 8 \\ 4.92 & - 19.205 & 28.8 \\ 16.8 & - 12.34 & 15.875 \end{array} | - 13 | \begin{array}{c} 16 & 8.915 & - 8 \\ 4.92 & 20.84 & 28.8 \\ 16.8 & 48.24 & 15.875 \end{array} | - 20.8 | \begin{array}{c} 16 & 8.915 & 20 \\ 4.92 & 20.84 & - 19.205 \\ 16.8 & 48.24 & - 12.34 \end{array} |$

Step 8.3

Evaluate $| \begin{array}{c} 20.84 & 28.8 \\ 48.24 & 15.875 \end{array} |$ .

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Step 8.3.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$24.915 (8.915 \cdot 50.512625 - 20 (20.84 \cdot 15.875 - 48.24 \cdot 28.8) - 8 | \begin{array}{c} 20.84 & - 19.205 \\ 48.24 & - 12.34 \end{array} |) - 8.48 | \begin{array}{c} 16 & 20 & - 8 \\ 4.92 & - 19.205 & 28.8 \\ 16.8 & - 12.34 & 15.875 \end{array} | - 13 | \begin{array}{c} 16 & 8.915 & - 8 \\ 4.92 & 20.84 & 28.8 \\ 16.8 & 48.24 & 15.875 \end{array} | - 20.8 | \begin{array}{c} 16 & 8.915 & 20 \\ 4.92 & 20.84 & - 19.205 \\ 16.8 & 48.24 & - 12.34 \end{array} |$

Step 8.3.2

Simplify the determinant.

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Step 8.3.2.1

Simplify each term.

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Step 8.3.2.1.1

Multiply $20.84$ by $15.875$ .

$24.915 (8.915 \cdot 50.512625 - 20 (330.835 - 48.24 \cdot 28.8) - 8 | \begin{array}{c} 20.84 & - 19.205 \\ 48.24 & - 12.34 \end{array} |) - 8.48 | \begin{array}{c} 16 & 20 & - 8 \\ 4.92 & - 19.205 & 28.8 \\ 16.8 & - 12.34 & 15.875 \end{array} | - 13 | \begin{array}{c} 16 & 8.915 & - 8 \\ 4.92 & 20.84 & 28.8 \\ 16.8 & 48.24 & 15.875 \end{array} | - 20.8 | \begin{array}{c} 16 & 8.915 & 20 \\ 4.92 & 20.84 & - 19.205 \\ 16.8 & 48.24 & - 12.34 \end{array} |$

Step 8.3.2.1.2

Multiply $- 48.24$ by $28.8$ .

$24.915 (8.915 \cdot 50.512625 - 20 (330.835 - 1389.312) - 8 | \begin{array}{c} 20.84 & - 19.205 \\ 48.24 & - 12.34 \end{array} |) - 8.48 | \begin{array}{c} 16 & 20 & - 8 \\ 4.92 & - 19.205 & 28.8 \\ 16.8 & - 12.34 & 15.875 \end{array} | - 13 | \begin{array}{c} 16 & 8.915 & - 8 \\ 4.92 & 20.84 & 28.8 \\ 16.8 & 48.24 & 15.875 \end{array} | - 20.8 | \begin{array}{c} 16 & 8.915 & 20 \\ 4.92 & 20.84 & - 19.205 \\ 16.8 & 48.24 & - 12.34 \end{array} |$

Step 8.3.2.2

Subtract $1389.312$ from $330.835$ .

$24.915 (8.915 \cdot 50.512625 - 20 \cdot - 1058.477 - 8 | \begin{array}{c} 20.84 & - 19.205 \\ 48.24 & - 12.34 \end{array} |) - 8.48 | \begin{array}{c} 16 & 20 & - 8 \\ 4.92 & - 19.205 & 28.8 \\ 16.8 & - 12.34 & 15.875 \end{array} | - 13 | \begin{array}{c} 16 & 8.915 & - 8 \\ 4.92 & 20.84 & 28.8 \\ 16.8 & 48.24 & 15.875 \end{array} | - 20.8 | \begin{array}{c} 16 & 8.915 & 20 \\ 4.92 & 20.84 & - 19.205 \\ 16.8 & 48.24 & - 12.34 \end{array} |$

Step 8.4

Evaluate $| \begin{array}{c} 20.84 & - 19.205 \\ 48.24 & - 12.34 \end{array} |$ .

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Step 8.4.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$24.915 (8.915 \cdot 50.512625 - 20 \cdot - 1058.477 - 8 (20.84 \cdot - 12.34 - 48.24 \cdot - 19.205)) - 8.48 | \begin{array}{c} 16 & 20 & - 8 \\ 4.92 & - 19.205 & 28.8 \\ 16.8 & - 12.34 & 15.875 \end{array} | - 13 | \begin{array}{c} 16 & 8.915 & - 8 \\ 4.92 & 20.84 & 28.8 \\ 16.8 & 48.24 & 15.875 \end{array} | - 20.8 | \begin{array}{c} 16 & 8.915 & 20 \\ 4.92 & 20.84 & - 19.205 \\ 16.8 & 48.24 & - 12.34 \end{array} |$

Step 8.4.2

Simplify the determinant.

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Step 8.4.2.1

Simplify each term.

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Step 8.4.2.1.1

Multiply $20.84$ by $- 12.34$ .

$24.915 (8.915 \cdot 50.512625 - 20 \cdot - 1058.477 - 8 (- 257.1656 - 48.24 \cdot - 19.205)) - 8.48 | \begin{array}{c} 16 & 20 & - 8 \\ 4.92 & - 19.205 & 28.8 \\ 16.8 & - 12.34 & 15.875 \end{array} | - 13 | \begin{array}{c} 16 & 8.915 & - 8 \\ 4.92 & 20.84 & 28.8 \\ 16.8 & 48.24 & 15.875 \end{array} | - 20.8 | \begin{array}{c} 16 & 8.915 & 20 \\ 4.92 & 20.84 & - 19.205 \\ 16.8 & 48.24 & - 12.34 \end{array} |$

Step 8.4.2.1.2

Multiply $- 48.24$ by $- 19.205$ .

$24.915 (8.915 \cdot 50.512625 - 20 \cdot - 1058.477 - 8 (- 257.1656 + 926.4492)) - 8.48 | \begin{array}{c} 16 & 20 & - 8 \\ 4.92 & - 19.205 & 28.8 \\ 16.8 & - 12.34 & 15.875 \end{array} | - 13 | \begin{array}{c} 16 & 8.915 & - 8 \\ 4.92 & 20.84 & 28.8 \\ 16.8 & 48.24 & 15.875 \end{array} | - 20.8 | \begin{array}{c} 16 & 8.915 & 20 \\ 4.92 & 20.84 & - 19.205 \\ 16.8 & 48.24 & - 12.34 \end{array} |$

Step 8.4.2.2

Add $- 257.1656$ and $926.4492$ .

$24.915 (8.915 \cdot 50.512625 - 20 \cdot - 1058.477 - 8 \cdot 669.2836) - 8.48 | \begin{array}{c} 16 & 20 & - 8 \\ 4.92 & - 19.205 & 28.8 \\ 16.8 & - 12.34 & 15.875 \end{array} | - 13 | \begin{array}{c} 16 & 8.915 & - 8 \\ 4.92 & 20.84 & 28.8 \\ 16.8 & 48.24 & 15.875 \end{array} | - 20.8 | \begin{array}{c} 16 & 8.915 & 20 \\ 4.92 & 20.84 & - 19.205 \\ 16.8 & 48.24 & - 12.34 \end{array} |$

Step 8.5

Simplify the determinant.

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Step 8.5.1

Simplify each term.

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Step 8.5.1.1

Multiply $8.915$ by $50.512625$ .

$24.915 (450.32005187 - 20 \cdot - 1058.477 - 8 \cdot 669.2836) - 8.48 | \begin{array}{c} 16 & 20 & - 8 \\ 4.92 & - 19.205 & 28.8 \\ 16.8 & - 12.34 & 15.875 \end{array} | - 13 | \begin{array}{c} 16 & 8.915 & - 8 \\ 4.92 & 20.84 & 28.8 \\ 16.8 & 48.24 & 15.875 \end{array} | - 20.8 | \begin{array}{c} 16 & 8.915 & 20 \\ 4.92 & 20.84 & - 19.205 \\ 16.8 & 48.24 & - 12.34 \end{array} |$

Step 8.5.1.2

Multiply $- 20$ by $- 1058.477$ .

$24.915 (450.32005187 + 21169.54 - 8 \cdot 669.2836) - 8.48 | \begin{array}{c} 16 & 20 & - 8 \\ 4.92 & - 19.205 & 28.8 \\ 16.8 & - 12.34 & 15.875 \end{array} | - 13 | \begin{array}{c} 16 & 8.915 & - 8 \\ 4.92 & 20.84 & 28.8 \\ 16.8 & 48.24 & 15.875 \end{array} | - 20.8 | \begin{array}{c} 16 & 8.915 & 20 \\ 4.92 & 20.84 & - 19.205 \\ 16.8 & 48.24 & - 12.34 \end{array} |$

Step 8.5.1.3

Multiply $- 8$ by $669.2836$ .

$24.915 (450.32005187 + 21169.54 - 5354.2688) - 8.48 | \begin{array}{c} 16 & 20 & - 8 \\ 4.92 & - 19.205 & 28.8 \\ 16.8 & - 12.34 & 15.875 \end{array} | - 13 | \begin{array}{c} 16 & 8.915 & - 8 \\ 4.92 & 20.84 & 28.8 \\ 16.8 & 48.24 & 15.875 \end{array} | - 20.8 | \begin{array}{c} 16 & 8.915 & 20 \\ 4.92 & 20.84 & - 19.205 \\ 16.8 & 48.24 & - 12.34 \end{array} |$

Step 8.5.2

Add $450.32005187$ and $21169.54$ .

$24.915 (21619.86005187 - 5354.2688) - 8.48 | \begin{array}{c} 16 & 20 & - 8 \\ 4.92 & - 19.205 & 28.8 \\ 16.8 & - 12.34 & 15.875 \end{array} | - 13 | \begin{array}{c} 16 & 8.915 & - 8 \\ 4.92 & 20.84 & 28.8 \\ 16.8 & 48.24 & 15.875 \end{array} | - 20.8 | \begin{array}{c} 16 & 8.915 & 20 \\ 4.92 & 20.84 & - 19.205 \\ 16.8 & 48.24 & - 12.34 \end{array} |$

Step 8.5.3

Subtract $5354.2688$ from $21619.86005187$ .

$24.915 \cdot 16265.59125187 - 8.48 | \begin{array}{c} 16 & 20 & - 8 \\ 4.92 & - 19.205 & 28.8 \\ 16.8 & - 12.34 & 15.875 \end{array} | - 13 | \begin{array}{c} 16 & 8.915 & - 8 \\ 4.92 & 20.84 & 28.8 \\ 16.8 & 48.24 & 15.875 \end{array} | - 20.8 | \begin{array}{c} 16 & 8.915 & 20 \\ 4.92 & 20.84 & - 19.205 \\ 16.8 & 48.24 & - 12.34 \end{array} |$

Step 9

Evaluate $| \begin{array}{c} 16 & 20 & - 8 \\ 4.92 & - 19.205 & 28.8 \\ 16.8 & - 12.34 & 15.875 \end{array} |$ .

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Step 9.1

Choose the row or column with the most $0$ elements. If there are no $0$ elements choose any row or column. Multiply every element in row $1$ by its cofactor and add.

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Step 9.1.1

Consider the corresponding sign chart.

$| \begin{array}{c} + & - & + \\ - & + & - \\ + & - & + \end{array} |$

Step 9.1.2

The cofactor is the minor with the sign changed if the indices match a $-$ position on the sign chart.

Step 9.1.3

The minor for $a_{11}$ is the determinant with row $1$ and column $1$ deleted.

$| \begin{array}{c} - 19.205 & 28.8 \\ - 12.34 & 15.875 \end{array} |$

Step 9.1.4

Multiply element $a_{11}$ by its cofactor.

$16 | \begin{array}{c} - 19.205 & 28.8 \\ - 12.34 & 15.875 \end{array} |$

Step 9.1.5

The minor for $a_{12}$ is the determinant with row $1$ and column $2$ deleted.

$| \begin{array}{c} 4.92 & 28.8 \\ 16.8 & 15.875 \end{array} |$

Step 9.1.6

Multiply element $a_{12}$ by its cofactor.

$- 20 | \begin{array}{c} 4.92 & 28.8 \\ 16.8 & 15.875 \end{array} |$

Step 9.1.7

The minor for $a_{13}$ is the determinant with row $1$ and column $3$ deleted.

$| \begin{array}{c} 4.92 & - 19.205 \\ 16.8 & - 12.34 \end{array} |$

Step 9.1.8

Multiply element $a_{13}$ by its cofactor.

$- 8 | \begin{array}{c} 4.92 & - 19.205 \\ 16.8 & - 12.34 \end{array} |$

Step 9.1.9

Add the terms together.

$24.915 \cdot 16265.59125187 - 8.48 (16 | \begin{array}{c} - 19.205 & 28.8 \\ - 12.34 & 15.875 \end{array} | - 20 | \begin{array}{c} 4.92 & 28.8 \\ 16.8 & 15.875 \end{array} | - 8 | \begin{array}{c} 4.92 & - 19.205 \\ 16.8 & - 12.34 \end{array} |) - 13 | \begin{array}{c} 16 & 8.915 & - 8 \\ 4.92 & 20.84 & 28.8 \\ 16.8 & 48.24 & 15.875 \end{array} | - 20.8 | \begin{array}{c} 16 & 8.915 & 20 \\ 4.92 & 20.84 & - 19.205 \\ 16.8 & 48.24 & - 12.34 \end{array} |$

Step 9.2

Evaluate $| \begin{array}{c} - 19.205 & 28.8 \\ - 12.34 & 15.875 \end{array} |$ .

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Step 9.2.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$24.915 \cdot 16265.59125187 - 8.48 (16 (- 19.205 \cdot 15.875 - (- 12.34 \cdot 28.8)) - 20 | \begin{array}{c} 4.92 & 28.8 \\ 16.8 & 15.875 \end{array} | - 8 | \begin{array}{c} 4.92 & - 19.205 \\ 16.8 & - 12.34 \end{array} |) - 13 | \begin{array}{c} 16 & 8.915 & - 8 \\ 4.92 & 20.84 & 28.8 \\ 16.8 & 48.24 & 15.875 \end{array} | - 20.8 | \begin{array}{c} 16 & 8.915 & 20 \\ 4.92 & 20.84 & - 19.205 \\ 16.8 & 48.24 & - 12.34 \end{array} |$

Step 9.2.2

Simplify the determinant.

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Step 9.2.2.1

Simplify each term.

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Step 9.2.2.1.1

Multiply $- 19.205$ by $15.875$ .

$24.915 \cdot 16265.59125187 - 8.48 (16 (- 304.879375 - (- 12.34 \cdot 28.8)) - 20 | \begin{array}{c} 4.92 & 28.8 \\ 16.8 & 15.875 \end{array} | - 8 | \begin{array}{c} 4.92 & - 19.205 \\ 16.8 & - 12.34 \end{array} |) - 13 | \begin{array}{c} 16 & 8.915 & - 8 \\ 4.92 & 20.84 & 28.8 \\ 16.8 & 48.24 & 15.875 \end{array} | - 20.8 | \begin{array}{c} 16 & 8.915 & 20 \\ 4.92 & 20.84 & - 19.205 \\ 16.8 & 48.24 & - 12.34 \end{array} |$

Step 9.2.2.1.2

Multiply $- (- 12.34 \cdot 28.8)$ .

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Step 9.2.2.1.2.1

Multiply $- 12.34$ by $28.8$ .

$24.915 \cdot 16265.59125187 - 8.48 (16 (- 304.879375 - - 355.392) - 20 | \begin{array}{c} 4.92 & 28.8 \\ 16.8 & 15.875 \end{array} | - 8 | \begin{array}{c} 4.92 & - 19.205 \\ 16.8 & - 12.34 \end{array} |) - 13 | \begin{array}{c} 16 & 8.915 & - 8 \\ 4.92 & 20.84 & 28.8 \\ 16.8 & 48.24 & 15.875 \end{array} | - 20.8 | \begin{array}{c} 16 & 8.915 & 20 \\ 4.92 & 20.84 & - 19.205 \\ 16.8 & 48.24 & - 12.34 \end{array} |$

Step 9.2.2.1.2.2

Multiply $- 1$ by $- 355.392$ .

$24.915 \cdot 16265.59125187 - 8.48 (16 (- 304.879375 + 355.392) - 20 | \begin{array}{c} 4.92 & 28.8 \\ 16.8 & 15.875 \end{array} | - 8 | \begin{array}{c} 4.92 & - 19.205 \\ 16.8 & - 12.34 \end{array} |) - 13 | \begin{array}{c} 16 & 8.915 & - 8 \\ 4.92 & 20.84 & 28.8 \\ 16.8 & 48.24 & 15.875 \end{array} | - 20.8 | \begin{array}{c} 16 & 8.915 & 20 \\ 4.92 & 20.84 & - 19.205 \\ 16.8 & 48.24 & - 12.34 \end{array} |$

Step 9.2.2.2

Add $- 304.879375$ and $355.392$ .

$24.915 \cdot 16265.59125187 - 8.48 (16 \cdot 50.512625 - 20 | \begin{array}{c} 4.92 & 28.8 \\ 16.8 & 15.875 \end{array} | - 8 | \begin{array}{c} 4.92 & - 19.205 \\ 16.8 & - 12.34 \end{array} |) - 13 | \begin{array}{c} 16 & 8.915 & - 8 \\ 4.92 & 20.84 & 28.8 \\ 16.8 & 48.24 & 15.875 \end{array} | - 20.8 | \begin{array}{c} 16 & 8.915 & 20 \\ 4.92 & 20.84 & - 19.205 \\ 16.8 & 48.24 & - 12.34 \end{array} |$

Step 9.3

Evaluate $| \begin{array}{c} 4.92 & 28.8 \\ 16.8 & 15.875 \end{array} |$ .

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Step 9.3.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$24.915 \cdot 16265.59125187 - 8.48 (16 \cdot 50.512625 - 20 (4.92 \cdot 15.875 - 16.8 \cdot 28.8) - 8 | \begin{array}{c} 4.92 & - 19.205 \\ 16.8 & - 12.34 \end{array} |) - 13 | \begin{array}{c} 16 & 8.915 & - 8 \\ 4.92 & 20.84 & 28.8 \\ 16.8 & 48.24 & 15.875 \end{array} | - 20.8 | \begin{array}{c} 16 & 8.915 & 20 \\ 4.92 & 20.84 & - 19.205 \\ 16.8 & 48.24 & - 12.34 \end{array} |$

Step 9.3.2

Simplify the determinant.

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Step 9.3.2.1

Simplify each term.

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Step 9.3.2.1.1

Multiply $4.92$ by $15.875$ .

$24.915 \cdot 16265.59125187 - 8.48 (16 \cdot 50.512625 - 20 (78.105 - 16.8 \cdot 28.8) - 8 | \begin{array}{c} 4.92 & - 19.205 \\ 16.8 & - 12.34 \end{array} |) - 13 | \begin{array}{c} 16 & 8.915 & - 8 \\ 4.92 & 20.84 & 28.8 \\ 16.8 & 48.24 & 15.875 \end{array} | - 20.8 | \begin{array}{c} 16 & 8.915 & 20 \\ 4.92 & 20.84 & - 19.205 \\ 16.8 & 48.24 & - 12.34 \end{array} |$

Step 9.3.2.1.2

Multiply $- 16.8$ by $28.8$ .

$24.915 \cdot 16265.59125187 - 8.48 (16 \cdot 50.512625 - 20 (78.105 - 483.84) - 8 | \begin{array}{c} 4.92 & - 19.205 \\ 16.8 & - 12.34 \end{array} |) - 13 | \begin{array}{c} 16 & 8.915 & - 8 \\ 4.92 & 20.84 & 28.8 \\ 16.8 & 48.24 & 15.875 \end{array} | - 20.8 | \begin{array}{c} 16 & 8.915 & 20 \\ 4.92 & 20.84 & - 19.205 \\ 16.8 & 48.24 & - 12.34 \end{array} |$

Step 9.3.2.2

Subtract $483.84$ from $78.105$ .

$24.915 \cdot 16265.59125187 - 8.48 (16 \cdot 50.512625 - 20 \cdot - 405.735 - 8 | \begin{array}{c} 4.92 & - 19.205 \\ 16.8 & - 12.34 \end{array} |) - 13 | \begin{array}{c} 16 & 8.915 & - 8 \\ 4.92 & 20.84 & 28.8 \\ 16.8 & 48.24 & 15.875 \end{array} | - 20.8 | \begin{array}{c} 16 & 8.915 & 20 \\ 4.92 & 20.84 & - 19.205 \\ 16.8 & 48.24 & - 12.34 \end{array} |$

Step 9.4

Evaluate $| \begin{array}{c} 4.92 & - 19.205 \\ 16.8 & - 12.34 \end{array} |$ .

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Step 9.4.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$24.915 \cdot 16265.59125187 - 8.48 (16 \cdot 50.512625 - 20 \cdot - 405.735 - 8 (4.92 \cdot - 12.34 - 16.8 \cdot - 19.205)) - 13 | \begin{array}{c} 16 & 8.915 & - 8 \\ 4.92 & 20.84 & 28.8 \\ 16.8 & 48.24 & 15.875 \end{array} | - 20.8 | \begin{array}{c} 16 & 8.915 & 20 \\ 4.92 & 20.84 & - 19.205 \\ 16.8 & 48.24 & - 12.34 \end{array} |$

Step 9.4.2

Simplify the determinant.

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Step 9.4.2.1

Simplify each term.

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Step 9.4.2.1.1

Multiply $4.92$ by $- 12.34$ .

$24.915 \cdot 16265.59125187 - 8.48 (16 \cdot 50.512625 - 20 \cdot - 405.735 - 8 (- 60.7128 - 16.8 \cdot - 19.205)) - 13 | \begin{array}{c} 16 & 8.915 & - 8 \\ 4.92 & 20.84 & 28.8 \\ 16.8 & 48.24 & 15.875 \end{array} | - 20.8 | \begin{array}{c} 16 & 8.915 & 20 \\ 4.92 & 20.84 & - 19.205 \\ 16.8 & 48.24 & - 12.34 \end{array} |$

Step 9.4.2.1.2

Multiply $- 16.8$ by $- 19.205$ .

$24.915 \cdot 16265.59125187 - 8.48 (16 \cdot 50.512625 - 20 \cdot - 405.735 - 8 (- 60.7128 + 322.644)) - 13 | \begin{array}{c} 16 & 8.915 & - 8 \\ 4.92 & 20.84 & 28.8 \\ 16.8 & 48.24 & 15.875 \end{array} | - 20.8 | \begin{array}{c} 16 & 8.915 & 20 \\ 4.92 & 20.84 & - 19.205 \\ 16.8 & 48.24 & - 12.34 \end{array} |$

Step 9.4.2.2

Add $- 60.7128$ and $322.644$ .

$24.915 \cdot 16265.59125187 - 8.48 (16 \cdot 50.512625 - 20 \cdot - 405.735 - 8 \cdot 261.9312) - 13 | \begin{array}{c} 16 & 8.915 & - 8 \\ 4.92 & 20.84 & 28.8 \\ 16.8 & 48.24 & 15.875 \end{array} | - 20.8 | \begin{array}{c} 16 & 8.915 & 20 \\ 4.92 & 20.84 & - 19.205 \\ 16.8 & 48.24 & - 12.34 \end{array} |$

Step 9.5

Simplify the determinant.

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Step 9.5.1

Simplify each term.

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Step 9.5.1.1

Multiply $16$ by $50.512625$ .

$24.915 \cdot 16265.59125187 - 8.48 (808.202 - 20 \cdot - 405.735 - 8 \cdot 261.9312) - 13 | \begin{array}{c} 16 & 8.915 & - 8 \\ 4.92 & 20.84 & 28.8 \\ 16.8 & 48.24 & 15.875 \end{array} | - 20.8 | \begin{array}{c} 16 & 8.915 & 20 \\ 4.92 & 20.84 & - 19.205 \\ 16.8 & 48.24 & - 12.34 \end{array} |$

Step 9.5.1.2

Multiply $- 20$ by $- 405.735$ .

$24.915 \cdot 16265.59125187 - 8.48 (808.202 + 8114.7 - 8 \cdot 261.9312) - 13 | \begin{array}{c} 16 & 8.915 & - 8 \\ 4.92 & 20.84 & 28.8 \\ 16.8 & 48.24 & 15.875 \end{array} | - 20.8 | \begin{array}{c} 16 & 8.915 & 20 \\ 4.92 & 20.84 & - 19.205 \\ 16.8 & 48.24 & - 12.34 \end{array} |$

Step 9.5.1.3

Multiply $- 8$ by $261.9312$ .

$24.915 \cdot 16265.59125187 - 8.48 (808.202 + 8114.7 - 2095.4496) - 13 | \begin{array}{c} 16 & 8.915 & - 8 \\ 4.92 & 20.84 & 28.8 \\ 16.8 & 48.24 & 15.875 \end{array} | - 20.8 | \begin{array}{c} 16 & 8.915 & 20 \\ 4.92 & 20.84 & - 19.205 \\ 16.8 & 48.24 & - 12.34 \end{array} |$

Step 9.5.2

Add $808.202$ and $8114.7$ .

$24.915 \cdot 16265.59125187 - 8.48 (8922.902 - 2095.4496) - 13 | \begin{array}{c} 16 & 8.915 & - 8 \\ 4.92 & 20.84 & 28.8 \\ 16.8 & 48.24 & 15.875 \end{array} | - 20.8 | \begin{array}{c} 16 & 8.915 & 20 \\ 4.92 & 20.84 & - 19.205 \\ 16.8 & 48.24 & - 12.34 \end{array} |$

Step 9.5.3

Subtract $2095.4496$ from $8922.902$ .

$24.915 \cdot 16265.59125187 - 8.48 \cdot 6827.4524 - 13 | \begin{array}{c} 16 & 8.915 & - 8 \\ 4.92 & 20.84 & 28.8 \\ 16.8 & 48.24 & 15.875 \end{array} | - 20.8 | \begin{array}{c} 16 & 8.915 & 20 \\ 4.92 & 20.84 & - 19.205 \\ 16.8 & 48.24 & - 12.34 \end{array} |$

Step 10

Evaluate $| \begin{array}{c} 16 & 8.915 & - 8 \\ 4.92 & 20.84 & 28.8 \\ 16.8 & 48.24 & 15.875 \end{array} |$ .

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Step 10.1

Choose the row or column with the most $0$ elements. If there are no $0$ elements choose any row or column. Multiply every element in row $1$ by its cofactor and add.

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Step 10.1.1

Consider the corresponding sign chart.

$| \begin{array}{c} + & - & + \\ - & + & - \\ + & - & + \end{array} |$

Step 10.1.2

The cofactor is the minor with the sign changed if the indices match a $-$ position on the sign chart.

Step 10.1.3

The minor for $a_{11}$ is the determinant with row $1$ and column $1$ deleted.

$| \begin{array}{c} 20.84 & 28.8 \\ 48.24 & 15.875 \end{array} |$

Step 10.1.4

Multiply element $a_{11}$ by its cofactor.

$16 | \begin{array}{c} 20.84 & 28.8 \\ 48.24 & 15.875 \end{array} |$

Step 10.1.5

The minor for $a_{12}$ is the determinant with row $1$ and column $2$ deleted.

$| \begin{array}{c} 4.92 & 28.8 \\ 16.8 & 15.875 \end{array} |$

Step 10.1.6

Multiply element $a_{12}$ by its cofactor.

$- 8.915 | \begin{array}{c} 4.92 & 28.8 \\ 16.8 & 15.875 \end{array} |$

Step 10.1.7

The minor for $a_{13}$ is the determinant with row $1$ and column $3$ deleted.

$| \begin{array}{c} 4.92 & 20.84 \\ 16.8 & 48.24 \end{array} |$

Step 10.1.8

Multiply element $a_{13}$ by its cofactor.

$- 8 | \begin{array}{c} 4.92 & 20.84 \\ 16.8 & 48.24 \end{array} |$

Step 10.1.9

Add the terms together.

$24.915 \cdot 16265.59125187 - 8.48 \cdot 6827.4524 - 13 (16 | \begin{array}{c} 20.84 & 28.8 \\ 48.24 & 15.875 \end{array} | - 8.915 | \begin{array}{c} 4.92 & 28.8 \\ 16.8 & 15.875 \end{array} | - 8 | \begin{array}{c} 4.92 & 20.84 \\ 16.8 & 48.24 \end{array} |) - 20.8 | \begin{array}{c} 16 & 8.915 & 20 \\ 4.92 & 20.84 & - 19.205 \\ 16.8 & 48.24 & - 12.34 \end{array} |$

Step 10.2

Evaluate $| \begin{array}{c} 20.84 & 28.8 \\ 48.24 & 15.875 \end{array} |$ .

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Step 10.2.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$24.915 \cdot 16265.59125187 - 8.48 \cdot 6827.4524 - 13 (16 (20.84 \cdot 15.875 - 48.24 \cdot 28.8) - 8.915 | \begin{array}{c} 4.92 & 28.8 \\ 16.8 & 15.875 \end{array} | - 8 | \begin{array}{c} 4.92 & 20.84 \\ 16.8 & 48.24 \end{array} |) - 20.8 | \begin{array}{c} 16 & 8.915 & 20 \\ 4.92 & 20.84 & - 19.205 \\ 16.8 & 48.24 & - 12.34 \end{array} |$

Step 10.2.2

Simplify the determinant.

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Step 10.2.2.1

Simplify each term.

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Step 10.2.2.1.1

Multiply $20.84$ by $15.875$ .

$24.915 \cdot 16265.59125187 - 8.48 \cdot 6827.4524 - 13 (16 (330.835 - 48.24 \cdot 28.8) - 8.915 | \begin{array}{c} 4.92 & 28.8 \\ 16.8 & 15.875 \end{array} | - 8 | \begin{array}{c} 4.92 & 20.84 \\ 16.8 & 48.24 \end{array} |) - 20.8 | \begin{array}{c} 16 & 8.915 & 20 \\ 4.92 & 20.84 & - 19.205 \\ 16.8 & 48.24 & - 12.34 \end{array} |$

Step 10.2.2.1.2

Multiply $- 48.24$ by $28.8$ .

$24.915 \cdot 16265.59125187 - 8.48 \cdot 6827.4524 - 13 (16 (330.835 - 1389.312) - 8.915 | \begin{array}{c} 4.92 & 28.8 \\ 16.8 & 15.875 \end{array} | - 8 | \begin{array}{c} 4.92 & 20.84 \\ 16.8 & 48.24 \end{array} |) - 20.8 | \begin{array}{c} 16 & 8.915 & 20 \\ 4.92 & 20.84 & - 19.205 \\ 16.8 & 48.24 & - 12.34 \end{array} |$

Step 10.2.2.2

Subtract $1389.312$ from $330.835$ .

$24.915 \cdot 16265.59125187 - 8.48 \cdot 6827.4524 - 13 (16 \cdot - 1058.477 - 8.915 | \begin{array}{c} 4.92 & 28.8 \\ 16.8 & 15.875 \end{array} | - 8 | \begin{array}{c} 4.92 & 20.84 \\ 16.8 & 48.24 \end{array} |) - 20.8 | \begin{array}{c} 16 & 8.915 & 20 \\ 4.92 & 20.84 & - 19.205 \\ 16.8 & 48.24 & - 12.34 \end{array} |$

Step 10.3

Evaluate $| \begin{array}{c} 4.92 & 28.8 \\ 16.8 & 15.875 \end{array} |$ .

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Step 10.3.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$24.915 \cdot 16265.59125187 - 8.48 \cdot 6827.4524 - 13 (16 \cdot - 1058.477 - 8.915 (4.92 \cdot 15.875 - 16.8 \cdot 28.8) - 8 | \begin{array}{c} 4.92 & 20.84 \\ 16.8 & 48.24 \end{array} |) - 20.8 | \begin{array}{c} 16 & 8.915 & 20 \\ 4.92 & 20.84 & - 19.205 \\ 16.8 & 48.24 & - 12.34 \end{array} |$

Step 10.3.2

Simplify the determinant.

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Step 10.3.2.1

Simplify each term.

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Step 10.3.2.1.1

Multiply $4.92$ by $15.875$ .

$24.915 \cdot 16265.59125187 - 8.48 \cdot 6827.4524 - 13 (16 \cdot - 1058.477 - 8.915 (78.105 - 16.8 \cdot 28.8) - 8 | \begin{array}{c} 4.92 & 20.84 \\ 16.8 & 48.24 \end{array} |) - 20.8 | \begin{array}{c} 16 & 8.915 & 20 \\ 4.92 & 20.84 & - 19.205 \\ 16.8 & 48.24 & - 12.34 \end{array} |$

Step 10.3.2.1.2

Multiply $- 16.8$ by $28.8$ .

$24.915 \cdot 16265.59125187 - 8.48 \cdot 6827.4524 - 13 (16 \cdot - 1058.477 - 8.915 (78.105 - 483.84) - 8 | \begin{array}{c} 4.92 & 20.84 \\ 16.8 & 48.24 \end{array} |) - 20.8 | \begin{array}{c} 16 & 8.915 & 20 \\ 4.92 & 20.84 & - 19.205 \\ 16.8 & 48.24 & - 12.34 \end{array} |$

Step 10.3.2.2

Subtract $483.84$ from $78.105$ .

$24.915 \cdot 16265.59125187 - 8.48 \cdot 6827.4524 - 13 (16 \cdot - 1058.477 - 8.915 \cdot - 405.735 - 8 | \begin{array}{c} 4.92 & 20.84 \\ 16.8 & 48.24 \end{array} |) - 20.8 | \begin{array}{c} 16 & 8.915 & 20 \\ 4.92 & 20.84 & - 19.205 \\ 16.8 & 48.24 & - 12.34 \end{array} |$

Step 10.4

Evaluate $| \begin{array}{c} 4.92 & 20.84 \\ 16.8 & 48.24 \end{array} |$ .

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Step 10.4.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$24.915 \cdot 16265.59125187 - 8.48 \cdot 6827.4524 - 13 (16 \cdot - 1058.477 - 8.915 \cdot - 405.735 - 8 (4.92 \cdot 48.24 - 16.8 \cdot 20.84)) - 20.8 | \begin{array}{c} 16 & 8.915 & 20 \\ 4.92 & 20.84 & - 19.205 \\ 16.8 & 48.24 & - 12.34 \end{array} |$

Step 10.4.2

Simplify the determinant.

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Step 10.4.2.1

Simplify each term.

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Step 10.4.2.1.1

Multiply $4.92$ by $48.24$ .

$24.915 \cdot 16265.59125187 - 8.48 \cdot 6827.4524 - 13 (16 \cdot - 1058.477 - 8.915 \cdot - 405.735 - 8 (237.3408 - 16.8 \cdot 20.84)) - 20.8 | \begin{array}{c} 16 & 8.915 & 20 \\ 4.92 & 20.84 & - 19.205 \\ 16.8 & 48.24 & - 12.34 \end{array} |$

Step 10.4.2.1.2

Multiply $- 16.8$ by $20.84$ .

$24.915 \cdot 16265.59125187 - 8.48 \cdot 6827.4524 - 13 (16 \cdot - 1058.477 - 8.915 \cdot - 405.735 - 8 (237.3408 - 350.112)) - 20.8 | \begin{array}{c} 16 & 8.915 & 20 \\ 4.92 & 20.84 & - 19.205 \\ 16.8 & 48.24 & - 12.34 \end{array} |$

Step 10.4.2.2

Subtract $350.112$ from $237.3408$ .

$24.915 \cdot 16265.59125187 - 8.48 \cdot 6827.4524 - 13 (16 \cdot - 1058.477 - 8.915 \cdot - 405.735 - 8 \cdot - 112.7712) - 20.8 | \begin{array}{c} 16 & 8.915 & 20 \\ 4.92 & 20.84 & - 19.205 \\ 16.8 & 48.24 & - 12.34 \end{array} |$

Step 10.5

Simplify the determinant.

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Step 10.5.1

Simplify each term.

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Step 10.5.1.1

Multiply $16$ by $- 1058.477$ .

$24.915 \cdot 16265.59125187 - 8.48 \cdot 6827.4524 - 13 (- 16935.632 - 8.915 \cdot - 405.735 - 8 \cdot - 112.7712) - 20.8 | \begin{array}{c} 16 & 8.915 & 20 \\ 4.92 & 20.84 & - 19.205 \\ 16.8 & 48.24 & - 12.34 \end{array} |$

Step 10.5.1.2

Multiply $- 8.915$ by $- 405.735$ .

$24.915 \cdot 16265.59125187 - 8.48 \cdot 6827.4524 - 13 (- 16935.632 + 3617.127525 - 8 \cdot - 112.7712) - 20.8 | \begin{array}{c} 16 & 8.915 & 20 \\ 4.92 & 20.84 & - 19.205 \\ 16.8 & 48.24 & - 12.34 \end{array} |$

Step 10.5.1.3

Multiply $- 8$ by $- 112.7712$ .

$24.915 \cdot 16265.59125187 - 8.48 \cdot 6827.4524 - 13 (- 16935.632 + 3617.127525 + 902.1696) - 20.8 | \begin{array}{c} 16 & 8.915 & 20 \\ 4.92 & 20.84 & - 19.205 \\ 16.8 & 48.24 & - 12.34 \end{array} |$

Step 10.5.2

Add $- 16935.632$ and $3617.127525$ .

$24.915 \cdot 16265.59125187 - 8.48 \cdot 6827.4524 - 13 (- 13318.504475 + 902.1696) - 20.8 | \begin{array}{c} 16 & 8.915 & 20 \\ 4.92 & 20.84 & - 19.205 \\ 16.8 & 48.24 & - 12.34 \end{array} |$

Step 10.5.3

Add $- 13318.504475$ and $902.1696$ .

$24.915 \cdot 16265.59125187 - 8.48 \cdot 6827.4524 - 13 \cdot - 12416.334875 - 20.8 | \begin{array}{c} 16 & 8.915 & 20 \\ 4.92 & 20.84 & - 19.205 \\ 16.8 & 48.24 & - 12.34 \end{array} |$

Step 11

Evaluate $| \begin{array}{c} 16 & 8.915 & 20 \\ 4.92 & 20.84 & - 19.205 \\ 16.8 & 48.24 & - 12.34 \end{array} |$ .

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Step 11.1

Choose the row or column with the most $0$ elements. If there are no $0$ elements choose any row or column. Multiply every element in row $1$ by its cofactor and add.

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Step 11.1.1

Consider the corresponding sign chart.

$| \begin{array}{c} + & - & + \\ - & + & - \\ + & - & + \end{array} |$

Step 11.1.2

The cofactor is the minor with the sign changed if the indices match a $-$ position on the sign chart.

Step 11.1.3

The minor for $a_{11}$ is the determinant with row $1$ and column $1$ deleted.

$| \begin{array}{c} 20.84 & - 19.205 \\ 48.24 & - 12.34 \end{array} |$

Step 11.1.4

Multiply element $a_{11}$ by its cofactor.

$16 | \begin{array}{c} 20.84 & - 19.205 \\ 48.24 & - 12.34 \end{array} |$

Step 11.1.5

The minor for $a_{12}$ is the determinant with row $1$ and column $2$ deleted.

$| \begin{array}{c} 4.92 & - 19.205 \\ 16.8 & - 12.34 \end{array} |$

Step 11.1.6

Multiply element $a_{12}$ by its cofactor.

$- 8.915 | \begin{array}{c} 4.92 & - 19.205 \\ 16.8 & - 12.34 \end{array} |$

Step 11.1.7

The minor for $a_{13}$ is the determinant with row $1$ and column $3$ deleted.

$| \begin{array}{c} 4.92 & 20.84 \\ 16.8 & 48.24 \end{array} |$

Step 11.1.8

Multiply element $a_{13}$ by its cofactor.

$20 | \begin{array}{c} 4.92 & 20.84 \\ 16.8 & 48.24 \end{array} |$

Step 11.1.9

Add the terms together.

$24.915 \cdot 16265.59125187 - 8.48 \cdot 6827.4524 - 13 \cdot - 12416.334875 - 20.8 (16 | \begin{array}{c} 20.84 & - 19.205 \\ 48.24 & - 12.34 \end{array} | - 8.915 | \begin{array}{c} 4.92 & - 19.205 \\ 16.8 & - 12.34 \end{array} | + 20 | \begin{array}{c} 4.92 & 20.84 \\ 16.8 & 48.24 \end{array} |)$

Step 11.2

Evaluate $| \begin{array}{c} 20.84 & - 19.205 \\ 48.24 & - 12.34 \end{array} |$ .

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Step 11.2.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$24.915 \cdot 16265.59125187 - 8.48 \cdot 6827.4524 - 13 \cdot - 12416.334875 - 20.8 (16 (20.84 \cdot - 12.34 - 48.24 \cdot - 19.205) - 8.915 | \begin{array}{c} 4.92 & - 19.205 \\ 16.8 & - 12.34 \end{array} | + 20 | \begin{array}{c} 4.92 & 20.84 \\ 16.8 & 48.24 \end{array} |)$

Step 11.2.2

Simplify the determinant.

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Step 11.2.2.1

Simplify each term.

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Step 11.2.2.1.1

Multiply $20.84$ by $- 12.34$ .

$24.915 \cdot 16265.59125187 - 8.48 \cdot 6827.4524 - 13 \cdot - 12416.334875 - 20.8 (16 (- 257.1656 - 48.24 \cdot - 19.205) - 8.915 | \begin{array}{c} 4.92 & - 19.205 \\ 16.8 & - 12.34 \end{array} | + 20 | \begin{array}{c} 4.92 & 20.84 \\ 16.8 & 48.24 \end{array} |)$

Step 11.2.2.1.2

Multiply $- 48.24$ by $- 19.205$ .

$24.915 \cdot 16265.59125187 - 8.48 \cdot 6827.4524 - 13 \cdot - 12416.334875 - 20.8 (16 (- 257.1656 + 926.4492) - 8.915 | \begin{array}{c} 4.92 & - 19.205 \\ 16.8 & - 12.34 \end{array} | + 20 | \begin{array}{c} 4.92 & 20.84 \\ 16.8 & 48.24 \end{array} |)$

Step 11.2.2.2

Add $- 257.1656$ and $926.4492$ .

$24.915 \cdot 16265.59125187 - 8.48 \cdot 6827.4524 - 13 \cdot - 12416.334875 - 20.8 (16 \cdot 669.2836 - 8.915 | \begin{array}{c} 4.92 & - 19.205 \\ 16.8 & - 12.34 \end{array} | + 20 | \begin{array}{c} 4.92 & 20.84 \\ 16.8 & 48.24 \end{array} |)$

Step 11.3

Evaluate $| \begin{array}{c} 4.92 & - 19.205 \\ 16.8 & - 12.34 \end{array} |$ .

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Step 11.3.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$24.915 \cdot 16265.59125187 - 8.48 \cdot 6827.4524 - 13 \cdot - 12416.334875 - 20.8 (16 \cdot 669.2836 - 8.915 (4.92 \cdot - 12.34 - 16.8 \cdot - 19.205) + 20 | \begin{array}{c} 4.92 & 20.84 \\ 16.8 & 48.24 \end{array} |)$

Step 11.3.2

Simplify the determinant.

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Step 11.3.2.1

Simplify each term.

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Step 11.3.2.1.1

Multiply $4.92$ by $- 12.34$ .

$24.915 \cdot 16265.59125187 - 8.48 \cdot 6827.4524 - 13 \cdot - 12416.334875 - 20.8 (16 \cdot 669.2836 - 8.915 (- 60.7128 - 16.8 \cdot - 19.205) + 20 | \begin{array}{c} 4.92 & 20.84 \\ 16.8 & 48.24 \end{array} |)$

Step 11.3.2.1.2

Multiply $- 16.8$ by $- 19.205$ .

$24.915 \cdot 16265.59125187 - 8.48 \cdot 6827.4524 - 13 \cdot - 12416.334875 - 20.8 (16 \cdot 669.2836 - 8.915 (- 60.7128 + 322.644) + 20 | \begin{array}{c} 4.92 & 20.84 \\ 16.8 & 48.24 \end{array} |)$

Step 11.3.2.2

Add $- 60.7128$ and $322.644$ .

$24.915 \cdot 16265.59125187 - 8.48 \cdot 6827.4524 - 13 \cdot - 12416.334875 - 20.8 (16 \cdot 669.2836 - 8.915 \cdot 261.9312 + 20 | \begin{array}{c} 4.92 & 20.84 \\ 16.8 & 48.24 \end{array} |)$

Step 11.4

Evaluate $| \begin{array}{c} 4.92 & 20.84 \\ 16.8 & 48.24 \end{array} |$ .

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Step 11.4.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$24.915 \cdot 16265.59125187 - 8.48 \cdot 6827.4524 - 13 \cdot - 12416.334875 - 20.8 (16 \cdot 669.2836 - 8.915 \cdot 261.9312 + 20 (4.92 \cdot 48.24 - 16.8 \cdot 20.84))$

Step 11.4.2

Simplify the determinant.

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Step 11.4.2.1

Simplify each term.

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Step 11.4.2.1.1

Multiply $4.92$ by $48.24$ .

$24.915 \cdot 16265.59125187 - 8.48 \cdot 6827.4524 - 13 \cdot - 12416.334875 - 20.8 (16 \cdot 669.2836 - 8.915 \cdot 261.9312 + 20 (237.3408 - 16.8 \cdot 20.84))$

Step 11.4.2.1.2

Multiply $- 16.8$ by $20.84$ .

$24.915 \cdot 16265.59125187 - 8.48 \cdot 6827.4524 - 13 \cdot - 12416.334875 - 20.8 (16 \cdot 669.2836 - 8.915 \cdot 261.9312 + 20 (237.3408 - 350.112))$

Step 11.4.2.2

Subtract $350.112$ from $237.3408$ .

$24.915 \cdot 16265.59125187 - 8.48 \cdot 6827.4524 - 13 \cdot - 12416.334875 - 20.8 (16 \cdot 669.2836 - 8.915 \cdot 261.9312 + 20 \cdot - 112.7712)$

Step 11.5

Simplify the determinant.

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Step 11.5.1

Simplify each term.

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Step 11.5.1.1

Multiply $16$ by $669.2836$ .

$24.915 \cdot 16265.59125187 - 8.48 \cdot 6827.4524 - 13 \cdot - 12416.334875 - 20.8 (10708.5376 - 8.915 \cdot 261.9312 + 20 \cdot - 112.7712)$

Step 11.5.1.2

Multiply $- 8.915$ by $261.9312$ .

$24.915 \cdot 16265.59125187 - 8.48 \cdot 6827.4524 - 13 \cdot - 12416.334875 - 20.8 (10708.5376 - 2335.116648 + 20 \cdot - 112.7712)$

Step 11.5.1.3

Multiply $20$ by $- 112.7712$ .

$24.915 \cdot 16265.59125187 - 8.48 \cdot 6827.4524 - 13 \cdot - 12416.334875 - 20.8 (10708.5376 - 2335.116648 - 2255.424)$

Step 11.5.2

Subtract $2335.116648$ from $10708.5376$ .

$24.915 \cdot 16265.59125187 - 8.48 \cdot 6827.4524 - 13 \cdot - 12416.334875 - 20.8 (8373.420952 - 2255.424)$

Step 11.5.3

Subtract $2255.424$ from $8373.420952$ .

$24.915 \cdot 16265.59125187 - 8.48 \cdot 6827.4524 - 13 \cdot - 12416.334875 - 20.8 \cdot 6117.996952$

Step 12

Simplify the determinant.

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Step 12.1

Simplify each term.

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Step 12.1.1

Multiply $24.915$ by $16265.59125187$ .

$405257.20604046 - 8.48 \cdot 6827.4524 - 13 \cdot - 12416.334875 - 20.8 \cdot 6117.996952$

Step 12.1.2

Multiply $- 8.48$ by $6827.4524$ .

$405257.20604046 - 57896.796352 - 13 \cdot - 12416.334875 - 20.8 \cdot 6117.996952$

Step 12.1.3

Multiply $- 13$ by $- 12416.334875$ .

$405257.20604046 - 57896.796352 + 161412.353375 - 20.8 \cdot 6117.996952$

Step 12.1.4

Multiply $- 20.8$ by $6117.996952$ .

$405257.20604046 - 57896.796352 + 161412.353375 - 127254.3366016$

Step 12.2

Subtract $57896.796352$ from $405257.20604046$ .

$347360.40968846 + 161412.353375 - 127254.3366016$

Step 12.3

Add $347360.40968846$ and $161412.353375$ .

$508772.76306346 - 127254.3366016$

Step 12.4

Subtract $127254.3366016$ from $508772.76306346$ .

$381518.42646186$