Linear Algebra Examples

$- 2 [\begin{array}{c} 9 & 7 \\ - 4 & 2 \\ - 5 & - 9 \end{array}] + 3 [\begin{array}{c} - 5 & - 2 \\ - 6 & 9 \\ 8 & - 8 \end{array}]$

Step 1

Simplify each term.

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Step 1.1

Multiply $- 2$ by each element of the matrix.

$[\begin{array}{c} - 2 \cdot 9 & - 2 \cdot 7 \\ - 2 \cdot - 4 & - 2 \cdot 2 \\ - 2 \cdot - 5 & - 2 \cdot - 9 \end{array}] + 3 [\begin{array}{c} - 5 & - 2 \\ - 6 & 9 \\ 8 & - 8 \end{array}]$

Step 1.2

Simplify each element in the matrix.

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Step 1.2.1

Multiply $- 2$ by $9$ .

$[\begin{array}{c} - 18 & - 2 \cdot 7 \\ - 2 \cdot - 4 & - 2 \cdot 2 \\ - 2 \cdot - 5 & - 2 \cdot - 9 \end{array}] + 3 [\begin{array}{c} - 5 & - 2 \\ - 6 & 9 \\ 8 & - 8 \end{array}]$

Step 1.2.2

Multiply $- 2$ by $7$ .

$[\begin{array}{c} - 18 & - 14 \\ - 2 \cdot - 4 & - 2 \cdot 2 \\ - 2 \cdot - 5 & - 2 \cdot - 9 \end{array}] + 3 [\begin{array}{c} - 5 & - 2 \\ - 6 & 9 \\ 8 & - 8 \end{array}]$

Step 1.2.3

Multiply $- 2$ by $- 4$ .

$[\begin{array}{c} - 18 & - 14 \\ 8 & - 2 \cdot 2 \\ - 2 \cdot - 5 & - 2 \cdot - 9 \end{array}] + 3 [\begin{array}{c} - 5 & - 2 \\ - 6 & 9 \\ 8 & - 8 \end{array}]$

Step 1.2.4

Multiply $- 2$ by $2$ .

$[\begin{array}{c} - 18 & - 14 \\ 8 & - 4 \\ - 2 \cdot - 5 & - 2 \cdot - 9 \end{array}] + 3 [\begin{array}{c} - 5 & - 2 \\ - 6 & 9 \\ 8 & - 8 \end{array}]$

Step 1.2.5

Multiply $- 2$ by $- 5$ .

$[\begin{array}{c} - 18 & - 14 \\ 8 & - 4 \\ 10 & - 2 \cdot - 9 \end{array}] + 3 [\begin{array}{c} - 5 & - 2 \\ - 6 & 9 \\ 8 & - 8 \end{array}]$

Step 1.2.6

Multiply $- 2$ by $- 9$ .

$[\begin{array}{c} - 18 & - 14 \\ 8 & - 4 \\ 10 & 18 \end{array}] + 3 [\begin{array}{c} - 5 & - 2 \\ - 6 & 9 \\ 8 & - 8 \end{array}]$

Step 1.3

Multiply $3$ by each element of the matrix.

$[\begin{array}{c} - 18 & - 14 \\ 8 & - 4 \\ 10 & 18 \end{array}] + [\begin{array}{c} 3 \cdot - 5 & 3 \cdot - 2 \\ 3 \cdot - 6 & 3 \cdot 9 \\ 3 \cdot 8 & 3 \cdot - 8 \end{array}]$

Step 1.4

Simplify each element in the matrix.

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Step 1.4.1

Multiply $3$ by $- 5$ .

$[\begin{array}{c} - 18 & - 14 \\ 8 & - 4 \\ 10 & 18 \end{array}] + [\begin{array}{c} - 15 & 3 \cdot - 2 \\ 3 \cdot - 6 & 3 \cdot 9 \\ 3 \cdot 8 & 3 \cdot - 8 \end{array}]$

Step 1.4.2

Multiply $3$ by $- 2$ .

$[\begin{array}{c} - 18 & - 14 \\ 8 & - 4 \\ 10 & 18 \end{array}] + [\begin{array}{c} - 15 & - 6 \\ 3 \cdot - 6 & 3 \cdot 9 \\ 3 \cdot 8 & 3 \cdot - 8 \end{array}]$

Step 1.4.3

Multiply $3$ by $- 6$ .

$[\begin{array}{c} - 18 & - 14 \\ 8 & - 4 \\ 10 & 18 \end{array}] + [\begin{array}{c} - 15 & - 6 \\ - 18 & 3 \cdot 9 \\ 3 \cdot 8 & 3 \cdot - 8 \end{array}]$

Step 1.4.4

Multiply $3$ by $9$ .

$[\begin{array}{c} - 18 & - 14 \\ 8 & - 4 \\ 10 & 18 \end{array}] + [\begin{array}{c} - 15 & - 6 \\ - 18 & 27 \\ 3 \cdot 8 & 3 \cdot - 8 \end{array}]$

Step 1.4.5

Multiply $3$ by $8$ .

$[\begin{array}{c} - 18 & - 14 \\ 8 & - 4 \\ 10 & 18 \end{array}] + [\begin{array}{c} - 15 & - 6 \\ - 18 & 27 \\ 24 & 3 \cdot - 8 \end{array}]$

Step 1.4.6

Multiply $3$ by $- 8$ .

$[\begin{array}{c} - 18 & - 14 \\ 8 & - 4 \\ 10 & 18 \end{array}] + [\begin{array}{c} - 15 & - 6 \\ - 18 & 27 \\ 24 & - 24 \end{array}]$

Step 2

Add the corresponding elements.

$[\begin{array}{c} - 18 - 15 & - 14 - 6 \\ 8 - 18 & - 4 + 27 \\ 10 + 24 & 18 - 24 \end{array}]$

Step 3

Simplify each element.

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Step 3.1

Subtract $15$ from $- 18$ .

$[\begin{array}{c} - 33 & - 14 - 6 \\ 8 - 18 & - 4 + 27 \\ 10 + 24 & 18 - 24 \end{array}]$

Step 3.2

Subtract $6$ from $- 14$ .

$[\begin{array}{c} - 33 & - 20 \\ 8 - 18 & - 4 + 27 \\ 10 + 24 & 18 - 24 \end{array}]$

Step 3.3

Subtract $18$ from $8$ .

$[\begin{array}{c} - 33 & - 20 \\ - 10 & - 4 + 27 \\ 10 + 24 & 18 - 24 \end{array}]$

Step 3.4

Add $- 4$ and $27$ .

$[\begin{array}{c} - 33 & - 20 \\ - 10 & 23 \\ 10 + 24 & 18 - 24 \end{array}]$

Step 3.5

Add $10$ and $24$ .

$[\begin{array}{c} - 33 & - 20 \\ - 10 & 23 \\ 34 & 18 - 24 \end{array}]$

Step 3.6

Subtract $24$ from $18$ .

$[\begin{array}{c} - 33 & - 20 \\ - 10 & 23 \\ 34 & - 6 \end{array}]$