Linear Algebra Examples

$[\begin{array}{c} 34 & 23 & 49 \\ 15 & 3 & 24 \\ 19 & 20 & 25 \end{array}] - 3 \cdot [\begin{array}{c} 8 & 7 & 11 \\ 3 & 6 & 3 \\ 5 & 1 & 8 \end{array}] + [\begin{array}{c} 2 & 1 & 3 \\ 1 & - 1 & 2 \\ 1 & 2 & 1 \end{array}]$

Step 1

Simplify each term.

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Step 1.1

Multiply $- 3$ by each element of the matrix.

$[\begin{array}{c} 34 & 23 & 49 \\ 15 & 3 & 24 \\ 19 & 20 & 25 \end{array}] + [\begin{array}{c} - 3 \cdot 8 & - 3 \cdot 7 & - 3 \cdot 11 \\ - 3 \cdot 3 & - 3 \cdot 6 & - 3 \cdot 3 \\ - 3 \cdot 5 & - 3 \cdot 1 & - 3 \cdot 8 \end{array}] + [\begin{array}{c} 2 & 1 & 3 \\ 1 & - 1 & 2 \\ 1 & 2 & 1 \end{array}]$

Step 1.2

Simplify each element in the matrix.

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Step 1.2.1

Multiply $- 3$ by $8$ .

$[\begin{array}{c} 34 & 23 & 49 \\ 15 & 3 & 24 \\ 19 & 20 & 25 \end{array}] + [\begin{array}{c} - 24 & - 3 \cdot 7 & - 3 \cdot 11 \\ - 3 \cdot 3 & - 3 \cdot 6 & - 3 \cdot 3 \\ - 3 \cdot 5 & - 3 \cdot 1 & - 3 \cdot 8 \end{array}] + [\begin{array}{c} 2 & 1 & 3 \\ 1 & - 1 & 2 \\ 1 & 2 & 1 \end{array}]$

Step 1.2.2

Multiply $- 3$ by $7$ .

$[\begin{array}{c} 34 & 23 & 49 \\ 15 & 3 & 24 \\ 19 & 20 & 25 \end{array}] + [\begin{array}{c} - 24 & - 21 & - 3 \cdot 11 \\ - 3 \cdot 3 & - 3 \cdot 6 & - 3 \cdot 3 \\ - 3 \cdot 5 & - 3 \cdot 1 & - 3 \cdot 8 \end{array}] + [\begin{array}{c} 2 & 1 & 3 \\ 1 & - 1 & 2 \\ 1 & 2 & 1 \end{array}]$

Step 1.2.3

Multiply $- 3$ by $11$ .

$[\begin{array}{c} 34 & 23 & 49 \\ 15 & 3 & 24 \\ 19 & 20 & 25 \end{array}] + [\begin{array}{c} - 24 & - 21 & - 33 \\ - 3 \cdot 3 & - 3 \cdot 6 & - 3 \cdot 3 \\ - 3 \cdot 5 & - 3 \cdot 1 & - 3 \cdot 8 \end{array}] + [\begin{array}{c} 2 & 1 & 3 \\ 1 & - 1 & 2 \\ 1 & 2 & 1 \end{array}]$

Step 1.2.4

Multiply $- 3$ by $3$ .

$[\begin{array}{c} 34 & 23 & 49 \\ 15 & 3 & 24 \\ 19 & 20 & 25 \end{array}] + [\begin{array}{c} - 24 & - 21 & - 33 \\ - 9 & - 3 \cdot 6 & - 3 \cdot 3 \\ - 3 \cdot 5 & - 3 \cdot 1 & - 3 \cdot 8 \end{array}] + [\begin{array}{c} 2 & 1 & 3 \\ 1 & - 1 & 2 \\ 1 & 2 & 1 \end{array}]$

Step 1.2.5

Multiply $- 3$ by $6$ .

$[\begin{array}{c} 34 & 23 & 49 \\ 15 & 3 & 24 \\ 19 & 20 & 25 \end{array}] + [\begin{array}{c} - 24 & - 21 & - 33 \\ - 9 & - 18 & - 3 \cdot 3 \\ - 3 \cdot 5 & - 3 \cdot 1 & - 3 \cdot 8 \end{array}] + [\begin{array}{c} 2 & 1 & 3 \\ 1 & - 1 & 2 \\ 1 & 2 & 1 \end{array}]$

Step 1.2.6

Multiply $- 3$ by $3$ .

$[\begin{array}{c} 34 & 23 & 49 \\ 15 & 3 & 24 \\ 19 & 20 & 25 \end{array}] + [\begin{array}{c} - 24 & - 21 & - 33 \\ - 9 & - 18 & - 9 \\ - 3 \cdot 5 & - 3 \cdot 1 & - 3 \cdot 8 \end{array}] + [\begin{array}{c} 2 & 1 & 3 \\ 1 & - 1 & 2 \\ 1 & 2 & 1 \end{array}]$

Step 1.2.7

Multiply $- 3$ by $5$ .

$[\begin{array}{c} 34 & 23 & 49 \\ 15 & 3 & 24 \\ 19 & 20 & 25 \end{array}] + [\begin{array}{c} - 24 & - 21 & - 33 \\ - 9 & - 18 & - 9 \\ - 15 & - 3 \cdot 1 & - 3 \cdot 8 \end{array}] + [\begin{array}{c} 2 & 1 & 3 \\ 1 & - 1 & 2 \\ 1 & 2 & 1 \end{array}]$

Step 1.2.8

Multiply $- 3$ by $1$ .

$[\begin{array}{c} 34 & 23 & 49 \\ 15 & 3 & 24 \\ 19 & 20 & 25 \end{array}] + [\begin{array}{c} - 24 & - 21 & - 33 \\ - 9 & - 18 & - 9 \\ - 15 & - 3 & - 3 \cdot 8 \end{array}] + [\begin{array}{c} 2 & 1 & 3 \\ 1 & - 1 & 2 \\ 1 & 2 & 1 \end{array}]$

Step 1.2.9

Multiply $- 3$ by $8$ .

$[\begin{array}{c} 34 & 23 & 49 \\ 15 & 3 & 24 \\ 19 & 20 & 25 \end{array}] + [\begin{array}{c} - 24 & - 21 & - 33 \\ - 9 & - 18 & - 9 \\ - 15 & - 3 & - 24 \end{array}] + [\begin{array}{c} 2 & 1 & 3 \\ 1 & - 1 & 2 \\ 1 & 2 & 1 \end{array}]$

Step 2

Add the corresponding elements.

$[\begin{array}{c} 34 - 24 & 23 - 21 & 49 - 33 \\ 15 - 9 & 3 - 18 & 24 - 9 \\ 19 - 15 & 20 - 3 & 25 - 24 \end{array}] + [\begin{array}{c} 2 & 1 & 3 \\ 1 & - 1 & 2 \\ 1 & 2 & 1 \end{array}]$

Step 3

Simplify each element.

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Step 3.1

Subtract $24$ from $34$ .

$[\begin{array}{c} 10 & 23 - 21 & 49 - 33 \\ 15 - 9 & 3 - 18 & 24 - 9 \\ 19 - 15 & 20 - 3 & 25 - 24 \end{array}] + [\begin{array}{c} 2 & 1 & 3 \\ 1 & - 1 & 2 \\ 1 & 2 & 1 \end{array}]$

Step 3.2

Subtract $21$ from $23$ .

$[\begin{array}{c} 10 & 2 & 49 - 33 \\ 15 - 9 & 3 - 18 & 24 - 9 \\ 19 - 15 & 20 - 3 & 25 - 24 \end{array}] + [\begin{array}{c} 2 & 1 & 3 \\ 1 & - 1 & 2 \\ 1 & 2 & 1 \end{array}]$

Step 3.3

Subtract $33$ from $49$ .

$[\begin{array}{c} 10 & 2 & 16 \\ 15 - 9 & 3 - 18 & 24 - 9 \\ 19 - 15 & 20 - 3 & 25 - 24 \end{array}] + [\begin{array}{c} 2 & 1 & 3 \\ 1 & - 1 & 2 \\ 1 & 2 & 1 \end{array}]$

Step 3.4

Subtract $9$ from $15$ .

$[\begin{array}{c} 10 & 2 & 16 \\ 6 & 3 - 18 & 24 - 9 \\ 19 - 15 & 20 - 3 & 25 - 24 \end{array}] + [\begin{array}{c} 2 & 1 & 3 \\ 1 & - 1 & 2 \\ 1 & 2 & 1 \end{array}]$

Step 3.5

Subtract $18$ from $3$ .

$[\begin{array}{c} 10 & 2 & 16 \\ 6 & - 15 & 24 - 9 \\ 19 - 15 & 20 - 3 & 25 - 24 \end{array}] + [\begin{array}{c} 2 & 1 & 3 \\ 1 & - 1 & 2 \\ 1 & 2 & 1 \end{array}]$

Step 3.6

Subtract $9$ from $24$ .

$[\begin{array}{c} 10 & 2 & 16 \\ 6 & - 15 & 15 \\ 19 - 15 & 20 - 3 & 25 - 24 \end{array}] + [\begin{array}{c} 2 & 1 & 3 \\ 1 & - 1 & 2 \\ 1 & 2 & 1 \end{array}]$

Step 3.7

Subtract $15$ from $19$ .

$[\begin{array}{c} 10 & 2 & 16 \\ 6 & - 15 & 15 \\ 4 & 20 - 3 & 25 - 24 \end{array}] + [\begin{array}{c} 2 & 1 & 3 \\ 1 & - 1 & 2 \\ 1 & 2 & 1 \end{array}]$

Step 3.8

Subtract $3$ from $20$ .

$[\begin{array}{c} 10 & 2 & 16 \\ 6 & - 15 & 15 \\ 4 & 17 & 25 - 24 \end{array}] + [\begin{array}{c} 2 & 1 & 3 \\ 1 & - 1 & 2 \\ 1 & 2 & 1 \end{array}]$

Step 3.9

Subtract $24$ from $25$ .

$[\begin{array}{c} 10 & 2 & 16 \\ 6 & - 15 & 15 \\ 4 & 17 & 1 \end{array}] + [\begin{array}{c} 2 & 1 & 3 \\ 1 & - 1 & 2 \\ 1 & 2 & 1 \end{array}]$

Step 4

Add the corresponding elements.

$[\begin{array}{c} 10 + 2 & 2 + 1 & 16 + 3 \\ 6 + 1 & - 15 - 1 & 15 + 2 \\ 4 + 1 & 17 + 2 & 1 + 1 \end{array}]$

Step 5

Simplify each element.

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Step 5.1

Add $10$ and $2$ .

$[\begin{array}{c} 12 & 2 + 1 & 16 + 3 \\ 6 + 1 & - 15 - 1 & 15 + 2 \\ 4 + 1 & 17 + 2 & 1 + 1 \end{array}]$

Step 5.2

Add $2$ and $1$ .

$[\begin{array}{c} 12 & 3 & 16 + 3 \\ 6 + 1 & - 15 - 1 & 15 + 2 \\ 4 + 1 & 17 + 2 & 1 + 1 \end{array}]$

Step 5.3

Add $16$ and $3$ .

$[\begin{array}{c} 12 & 3 & 19 \\ 6 + 1 & - 15 - 1 & 15 + 2 \\ 4 + 1 & 17 + 2 & 1 + 1 \end{array}]$

Step 5.4

Add $6$ and $1$ .

$[\begin{array}{c} 12 & 3 & 19 \\ 7 & - 15 - 1 & 15 + 2 \\ 4 + 1 & 17 + 2 & 1 + 1 \end{array}]$

Step 5.5

Subtract $1$ from $- 15$ .

$[\begin{array}{c} 12 & 3 & 19 \\ 7 & - 16 & 15 + 2 \\ 4 + 1 & 17 + 2 & 1 + 1 \end{array}]$

Step 5.6

Add $15$ and $2$ .

$[\begin{array}{c} 12 & 3 & 19 \\ 7 & - 16 & 17 \\ 4 + 1 & 17 + 2 & 1 + 1 \end{array}]$

Step 5.7

Add $4$ and $1$ .

$[\begin{array}{c} 12 & 3 & 19 \\ 7 & - 16 & 17 \\ 5 & 17 + 2 & 1 + 1 \end{array}]$

Step 5.8

Add $17$ and $2$ .

$[\begin{array}{c} 12 & 3 & 19 \\ 7 & - 16 & 17 \\ 5 & 19 & 1 + 1 \end{array}]$

Step 5.9

Add $1$ and $1$ .

$[\begin{array}{c} 12 & 3 & 19 \\ 7 & - 16 & 17 \\ 5 & 19 & 2 \end{array}]$

Step 6

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 6.1

Consider the corresponding sign chart.

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

Step 6.2

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

Step 6.3

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

$| \begin{array}{c} - 16 & 17 \\ 19 & 2 \end{array} |$

Step 6.4

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

$12 | \begin{array}{c} - 16 & 17 \\ 19 & 2 \end{array} |$

Step 6.5

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

$| \begin{array}{c} 7 & 17 \\ 5 & 2 \end{array} |$

Step 6.6

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

$- 3 | \begin{array}{c} 7 & 17 \\ 5 & 2 \end{array} |$

Step 6.7

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

$| \begin{array}{c} 7 & - 16 \\ 5 & 19 \end{array} |$

Step 6.8

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

$19 | \begin{array}{c} 7 & - 16 \\ 5 & 19 \end{array} |$

Step 6.9

Add the terms together.

$12 | \begin{array}{c} - 16 & 17 \\ 19 & 2 \end{array} | - 3 | \begin{array}{c} 7 & 17 \\ 5 & 2 \end{array} | + 19 | \begin{array}{c} 7 & - 16 \\ 5 & 19 \end{array} |$

Step 7

Evaluate $| \begin{array}{c} - 16 & 17 \\ 19 & 2 \end{array} |$ .

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Step 7.1

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

$12 (- 16 \cdot 2 - 19 \cdot 17) - 3 | \begin{array}{c} 7 & 17 \\ 5 & 2 \end{array} | + 19 | \begin{array}{c} 7 & - 16 \\ 5 & 19 \end{array} |$

Step 7.2

Simplify the determinant.

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Step 7.2.1

Simplify each term.

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Step 7.2.1.1

Multiply $- 16$ by $2$ .

$12 (- 32 - 19 \cdot 17) - 3 | \begin{array}{c} 7 & 17 \\ 5 & 2 \end{array} | + 19 | \begin{array}{c} 7 & - 16 \\ 5 & 19 \end{array} |$

Step 7.2.1.2

Multiply $- 19$ by $17$ .

$12 (- 32 - 323) - 3 | \begin{array}{c} 7 & 17 \\ 5 & 2 \end{array} | + 19 | \begin{array}{c} 7 & - 16 \\ 5 & 19 \end{array} |$

Step 7.2.2

Subtract $323$ from $- 32$ .

$12 \cdot - 355 - 3 | \begin{array}{c} 7 & 17 \\ 5 & 2 \end{array} | + 19 | \begin{array}{c} 7 & - 16 \\ 5 & 19 \end{array} |$

Step 8

Evaluate $| \begin{array}{c} 7 & 17 \\ 5 & 2 \end{array} |$ .

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Step 8.1

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

$12 \cdot - 355 - 3 (7 \cdot 2 - 5 \cdot 17) + 19 | \begin{array}{c} 7 & - 16 \\ 5 & 19 \end{array} |$

Step 8.2

Simplify the determinant.

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Step 8.2.1

Simplify each term.

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Step 8.2.1.1

Multiply $7$ by $2$ .

$12 \cdot - 355 - 3 (14 - 5 \cdot 17) + 19 | \begin{array}{c} 7 & - 16 \\ 5 & 19 \end{array} |$

Step 8.2.1.2

Multiply $- 5$ by $17$ .

$12 \cdot - 355 - 3 (14 - 85) + 19 | \begin{array}{c} 7 & - 16 \\ 5 & 19 \end{array} |$

Step 8.2.2

Subtract $85$ from $14$ .

$12 \cdot - 355 - 3 \cdot - 71 + 19 | \begin{array}{c} 7 & - 16 \\ 5 & 19 \end{array} |$

Step 9

Evaluate $| \begin{array}{c} 7 & - 16 \\ 5 & 19 \end{array} |$ .

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Step 9.1

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

$12 \cdot - 355 - 3 \cdot - 71 + 19 (7 \cdot 19 - 5 \cdot - 16)$

Step 9.2

Simplify the determinant.

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Step 9.2.1

Simplify each term.

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Step 9.2.1.1

Multiply $7$ by $19$ .

$12 \cdot - 355 - 3 \cdot - 71 + 19 (133 - 5 \cdot - 16)$

Step 9.2.1.2

Multiply $- 5$ by $- 16$ .

$12 \cdot - 355 - 3 \cdot - 71 + 19 (133 + 80)$

Step 9.2.2

Add $133$ and $80$ .

$12 \cdot - 355 - 3 \cdot - 71 + 19 \cdot 213$

Step 10

Simplify the determinant.

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Step 10.1

Simplify each term.

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Step 10.1.1

Multiply $12$ by $- 355$ .

$- 4260 - 3 \cdot - 71 + 19 \cdot 213$

Step 10.1.2

Multiply $- 3$ by $- 71$ .

$- 4260 + 213 + 19 \cdot 213$

Step 10.1.3

Multiply $19$ by $213$ .

$- 4260 + 213 + 4047$

Step 10.2

Add $- 4260$ and $213$ .

$- 4047 + 4047$

Step 10.3

Add $- 4047$ and $4047$ .

$0$