Linear Algebra Examples

$3 [\begin{array}{c} 1 & 3 \\ 4 & 2 \end{array}] - [\begin{array}{c} 3 & 3 \\ 4 & 4 \end{array}]$

Step 1

Simplify each term.

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Step 1.1

Multiply $3$ by each element of the matrix.

$[\begin{array}{c} 3 \cdot 1 & 3 \cdot 3 \\ 3 \cdot 4 & 3 \cdot 2 \end{array}] - [\begin{array}{c} 3 & 3 \\ 4 & 4 \end{array}]$

Step 1.2

Simplify each element in the matrix.

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Step 1.2.1

Multiply $3$ by $1$ .

$[\begin{array}{c} 3 & 3 \cdot 3 \\ 3 \cdot 4 & 3 \cdot 2 \end{array}] - [\begin{array}{c} 3 & 3 \\ 4 & 4 \end{array}]$

Step 1.2.2

Multiply $3$ by $3$ .

$[\begin{array}{c} 3 & 9 \\ 3 \cdot 4 & 3 \cdot 2 \end{array}] - [\begin{array}{c} 3 & 3 \\ 4 & 4 \end{array}]$

Step 1.2.3

Multiply $3$ by $4$ .

$[\begin{array}{c} 3 & 9 \\ 12 & 3 \cdot 2 \end{array}] - [\begin{array}{c} 3 & 3 \\ 4 & 4 \end{array}]$

Step 1.2.4

Multiply $3$ by $2$ .

$[\begin{array}{c} 3 & 9 \\ 12 & 6 \end{array}] - [\begin{array}{c} 3 & 3 \\ 4 & 4 \end{array}]$

Step 2

Subtract the corresponding elements.

$[\begin{array}{c} 3 - 3 & 9 - 3 \\ 12 - 4 & 6 - 4 \end{array}]$

Step 3

Simplify each element.

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Step 3.1

Subtract $3$ from $3$ .

$[\begin{array}{c} 0 & 9 - 3 \\ 12 - 4 & 6 - 4 \end{array}]$

Step 3.2

Subtract $3$ from $9$ .

$[\begin{array}{c} 0 & 6 \\ 12 - 4 & 6 - 4 \end{array}]$

Step 3.3

Subtract $4$ from $12$ .

$[\begin{array}{c} 0 & 6 \\ 8 & 6 - 4 \end{array}]$

Step 3.4

Subtract $4$ from $6$ .

$[\begin{array}{c} 0 & 6 \\ 8 & 2 \end{array}]$

Step 4

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

$0 \cdot 2 - 8 \cdot 6$

Step 5

Simplify the determinant.

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Step 5.1

Simplify each term.

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Step 5.1.1

Multiply $0$ by $2$ .

$0 - 8 \cdot 6$

Step 5.1.2

Multiply $- 8$ by $6$ .

$0 - 48$

Step 5.2

Subtract $48$ from $0$ .

$- 48$