Linear Algebra Examples

$[\begin{array}{c} 0.3 & 0.2 & 0.3 \\ 0.4 & 0.3 & 0.3 \\ 0.3 & 0.3 & 0.2 \end{array}] - [\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]$

Step 1

Subtract the corresponding elements.

$[\begin{array}{c} 0.3 - 1 & 0.2 - 0 & 0.3 - 0 \\ 0.4 - 0 & 0.3 - 1 & 0.3 - 0 \\ 0.3 - 0 & 0.3 - 0 & 0.2 - 1 \end{array}]$

Step 2

Simplify each element.

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Step 2.1

Subtract $1$ from $0.3$ .

$[\begin{array}{c} - 0.7 & 0.2 - 0 & 0.3 - 0 \\ 0.4 - 0 & 0.3 - 1 & 0.3 - 0 \\ 0.3 - 0 & 0.3 - 0 & 0.2 - 1 \end{array}]$

Step 2.2

Subtract $0$ from $0.2$ .

$[\begin{array}{c} - 0.7 & 0.2 & 0.3 - 0 \\ 0.4 - 0 & 0.3 - 1 & 0.3 - 0 \\ 0.3 - 0 & 0.3 - 0 & 0.2 - 1 \end{array}]$

Step 2.3

Subtract $0$ from $0.3$ .

$[\begin{array}{c} - 0.7 & 0.2 & 0.3 \\ 0.4 - 0 & 0.3 - 1 & 0.3 - 0 \\ 0.3 - 0 & 0.3 - 0 & 0.2 - 1 \end{array}]$

Step 2.4

Subtract $0$ from $0.4$ .

$[\begin{array}{c} - 0.7 & 0.2 & 0.3 \\ 0.4 & 0.3 - 1 & 0.3 - 0 \\ 0.3 - 0 & 0.3 - 0 & 0.2 - 1 \end{array}]$

Step 2.5

Subtract $1$ from $0.3$ .

$[\begin{array}{c} - 0.7 & 0.2 & 0.3 \\ 0.4 & - 0.7 & 0.3 - 0 \\ 0.3 - 0 & 0.3 - 0 & 0.2 - 1 \end{array}]$

Step 2.6

Subtract $0$ from $0.3$ .

$[\begin{array}{c} - 0.7 & 0.2 & 0.3 \\ 0.4 & - 0.7 & 0.3 \\ 0.3 - 0 & 0.3 - 0 & 0.2 - 1 \end{array}]$

Step 2.7

Subtract $0$ from $0.3$ .

$[\begin{array}{c} - 0.7 & 0.2 & 0.3 \\ 0.4 & - 0.7 & 0.3 \\ 0.3 & 0.3 - 0 & 0.2 - 1 \end{array}]$

Step 2.8

Subtract $0$ from $0.3$ .

$[\begin{array}{c} - 0.7 & 0.2 & 0.3 \\ 0.4 & - 0.7 & 0.3 \\ 0.3 & 0.3 & 0.2 - 1 \end{array}]$

Step 2.9

Subtract $1$ from $0.2$ .

$[\begin{array}{c} - 0.7 & 0.2 & 0.3 \\ 0.4 & - 0.7 & 0.3 \\ 0.3 & 0.3 & - 0.8 \end{array}]$

Step 3

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 3.1

Consider the corresponding sign chart.

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

Step 3.2

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

Step 3.3

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

$| \begin{array}{c} - 0.7 & 0.3 \\ 0.3 & - 0.8 \end{array} |$

Step 3.4

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

$- 0.7 | \begin{array}{c} - 0.7 & 0.3 \\ 0.3 & - 0.8 \end{array} |$

Step 3.5

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

$| \begin{array}{c} 0.4 & 0.3 \\ 0.3 & - 0.8 \end{array} |$

Step 3.6

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

$- 0.2 | \begin{array}{c} 0.4 & 0.3 \\ 0.3 & - 0.8 \end{array} |$

Step 3.7

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

$| \begin{array}{c} 0.4 & - 0.7 \\ 0.3 & 0.3 \end{array} |$

Step 3.8

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

$0.3 | \begin{array}{c} 0.4 & - 0.7 \\ 0.3 & 0.3 \end{array} |$

Step 3.9

Add the terms together.

$- 0.7 | \begin{array}{c} - 0.7 & 0.3 \\ 0.3 & - 0.8 \end{array} | - 0.2 | \begin{array}{c} 0.4 & 0.3 \\ 0.3 & - 0.8 \end{array} | + 0.3 | \begin{array}{c} 0.4 & - 0.7 \\ 0.3 & 0.3 \end{array} |$

Step 4

Evaluate $| \begin{array}{c} - 0.7 & 0.3 \\ 0.3 & - 0.8 \end{array} |$ .

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Step 4.1

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

$- 0.7 (- 0.7 \cdot - 0.8 - 0.3 \cdot 0.3) - 0.2 | \begin{array}{c} 0.4 & 0.3 \\ 0.3 & - 0.8 \end{array} | + 0.3 | \begin{array}{c} 0.4 & - 0.7 \\ 0.3 & 0.3 \end{array} |$

Step 4.2

Simplify the determinant.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Multiply $- 0.7$ by $- 0.8$ .

$- 0.7 (0.56 - 0.3 \cdot 0.3) - 0.2 | \begin{array}{c} 0.4 & 0.3 \\ 0.3 & - 0.8 \end{array} | + 0.3 | \begin{array}{c} 0.4 & - 0.7 \\ 0.3 & 0.3 \end{array} |$

Step 4.2.1.2

Multiply $- 0.3$ by $0.3$ .

$- 0.7 (0.56 - 0.09) - 0.2 | \begin{array}{c} 0.4 & 0.3 \\ 0.3 & - 0.8 \end{array} | + 0.3 | \begin{array}{c} 0.4 & - 0.7 \\ 0.3 & 0.3 \end{array} |$

Step 4.2.2

Subtract $0.09$ from $0.56$ .

$- 0.7 \cdot 0.47 - 0.2 | \begin{array}{c} 0.4 & 0.3 \\ 0.3 & - 0.8 \end{array} | + 0.3 | \begin{array}{c} 0.4 & - 0.7 \\ 0.3 & 0.3 \end{array} |$

Step 5

Evaluate $| \begin{array}{c} 0.4 & 0.3 \\ 0.3 & - 0.8 \end{array} |$ .

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Step 5.1

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

$- 0.7 \cdot 0.47 - 0.2 (0.4 \cdot - 0.8 - 0.3 \cdot 0.3) + 0.3 | \begin{array}{c} 0.4 & - 0.7 \\ 0.3 & 0.3 \end{array} |$

Step 5.2

Simplify the determinant.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

Multiply $0.4$ by $- 0.8$ .

$- 0.7 \cdot 0.47 - 0.2 (- 0.32 - 0.3 \cdot 0.3) + 0.3 | \begin{array}{c} 0.4 & - 0.7 \\ 0.3 & 0.3 \end{array} |$

Step 5.2.1.2

Multiply $- 0.3$ by $0.3$ .

$- 0.7 \cdot 0.47 - 0.2 (- 0.32 - 0.09) + 0.3 | \begin{array}{c} 0.4 & - 0.7 \\ 0.3 & 0.3 \end{array} |$

Step 5.2.2

Subtract $0.09$ from $- 0.32$ .

$- 0.7 \cdot 0.47 - 0.2 \cdot - 0.41 + 0.3 | \begin{array}{c} 0.4 & - 0.7 \\ 0.3 & 0.3 \end{array} |$

Step 6

Evaluate $| \begin{array}{c} 0.4 & - 0.7 \\ 0.3 & 0.3 \end{array} |$ .

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Step 6.1

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

$- 0.7 \cdot 0.47 - 0.2 \cdot - 0.41 + 0.3 (0.4 \cdot 0.3 - 0.3 \cdot - 0.7)$

Step 6.2

Simplify the determinant.

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

Multiply $0.4$ by $0.3$ .

$- 0.7 \cdot 0.47 - 0.2 \cdot - 0.41 + 0.3 (0.12 - 0.3 \cdot - 0.7)$

Step 6.2.1.2

Multiply $- 0.3$ by $- 0.7$ .

$- 0.7 \cdot 0.47 - 0.2 \cdot - 0.41 + 0.3 (0.12 + 0.21)$

Step 6.2.2

Add $0.12$ and $0.21$ .

$- 0.7 \cdot 0.47 - 0.2 \cdot - 0.41 + 0.3 \cdot 0.33$

Step 7

Simplify the determinant.

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Step 7.1

Simplify each term.

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Step 7.1.1

Multiply $- 0.7$ by $0.47$ .

$- 0.329 - 0.2 \cdot - 0.41 + 0.3 \cdot 0.33$

Step 7.1.2

Multiply $- 0.2$ by $- 0.41$ .

$- 0.329 + 0.082 + 0.3 \cdot 0.33$

Step 7.1.3

Multiply $0.3$ by $0.33$ .

$- 0.329 + 0.082 + 0.099$

Step 7.2

Add $- 0.329$ and $0.082$ .

$- 0.247 + 0.099$

Step 7.3

Add $- 0.247$ and $0.099$ .

$- 0.148$