Algebra Examples

$5 [\begin{array}{c} 1 & 4 \\ 2 & 3 \end{array}] + 6 [\begin{array}{c} 0 & 4 \\ - 1 & 3 \end{array}] + 2 [\begin{array}{c} 0 & 1 \\ - 1 & 2 \end{array}]$

Step 1

Simplify each term.

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Step 1.1

Multiply $5$ by each element of the matrix.

$[\begin{array}{c} 5 \cdot 1 & 5 \cdot 4 \\ 5 \cdot 2 & 5 \cdot 3 \end{array}] + 6 [\begin{array}{c} 0 & 4 \\ - 1 & 3 \end{array}] + 2 [\begin{array}{c} 0 & 1 \\ - 1 & 2 \end{array}]$

Step 1.2

Simplify each element in the matrix.

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Step 1.2.1

Multiply $5$ by $1$ .

$[\begin{array}{c} 5 & 5 \cdot 4 \\ 5 \cdot 2 & 5 \cdot 3 \end{array}] + 6 [\begin{array}{c} 0 & 4 \\ - 1 & 3 \end{array}] + 2 [\begin{array}{c} 0 & 1 \\ - 1 & 2 \end{array}]$

Step 1.2.2

Multiply $5$ by $4$ .

$[\begin{array}{c} 5 & 20 \\ 5 \cdot 2 & 5 \cdot 3 \end{array}] + 6 [\begin{array}{c} 0 & 4 \\ - 1 & 3 \end{array}] + 2 [\begin{array}{c} 0 & 1 \\ - 1 & 2 \end{array}]$

Step 1.2.3

Multiply $5$ by $2$ .

$[\begin{array}{c} 5 & 20 \\ 10 & 5 \cdot 3 \end{array}] + 6 [\begin{array}{c} 0 & 4 \\ - 1 & 3 \end{array}] + 2 [\begin{array}{c} 0 & 1 \\ - 1 & 2 \end{array}]$

Step 1.2.4

Multiply $5$ by $3$ .

$[\begin{array}{c} 5 & 20 \\ 10 & 15 \end{array}] + 6 [\begin{array}{c} 0 & 4 \\ - 1 & 3 \end{array}] + 2 [\begin{array}{c} 0 & 1 \\ - 1 & 2 \end{array}]$

Step 1.3

Multiply $6$ by each element of the matrix.

$[\begin{array}{c} 5 & 20 \\ 10 & 15 \end{array}] + [\begin{array}{c} 6 \cdot 0 & 6 \cdot 4 \\ 6 \cdot - 1 & 6 \cdot 3 \end{array}] + 2 [\begin{array}{c} 0 & 1 \\ - 1 & 2 \end{array}]$

Step 1.4

Simplify each element in the matrix.

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Step 1.4.1

Multiply $6$ by $0$ .

$[\begin{array}{c} 5 & 20 \\ 10 & 15 \end{array}] + [\begin{array}{c} 0 & 6 \cdot 4 \\ 6 \cdot - 1 & 6 \cdot 3 \end{array}] + 2 [\begin{array}{c} 0 & 1 \\ - 1 & 2 \end{array}]$

Step 1.4.2

Multiply $6$ by $4$ .

$[\begin{array}{c} 5 & 20 \\ 10 & 15 \end{array}] + [\begin{array}{c} 0 & 24 \\ 6 \cdot - 1 & 6 \cdot 3 \end{array}] + 2 [\begin{array}{c} 0 & 1 \\ - 1 & 2 \end{array}]$

Step 1.4.3

Multiply $6$ by $- 1$ .

$[\begin{array}{c} 5 & 20 \\ 10 & 15 \end{array}] + [\begin{array}{c} 0 & 24 \\ - 6 & 6 \cdot 3 \end{array}] + 2 [\begin{array}{c} 0 & 1 \\ - 1 & 2 \end{array}]$

Step 1.4.4

Multiply $6$ by $3$ .

$[\begin{array}{c} 5 & 20 \\ 10 & 15 \end{array}] + [\begin{array}{c} 0 & 24 \\ - 6 & 18 \end{array}] + 2 [\begin{array}{c} 0 & 1 \\ - 1 & 2 \end{array}]$

Step 1.5

Multiply $2$ by each element of the matrix.

$[\begin{array}{c} 5 & 20 \\ 10 & 15 \end{array}] + [\begin{array}{c} 0 & 24 \\ - 6 & 18 \end{array}] + [\begin{array}{c} 2 \cdot 0 & 2 \cdot 1 \\ 2 \cdot - 1 & 2 \cdot 2 \end{array}]$

Step 1.6

Simplify each element in the matrix.

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Step 1.6.1

Multiply $2$ by $0$ .

$[\begin{array}{c} 5 & 20 \\ 10 & 15 \end{array}] + [\begin{array}{c} 0 & 24 \\ - 6 & 18 \end{array}] + [\begin{array}{c} 0 & 2 \cdot 1 \\ 2 \cdot - 1 & 2 \cdot 2 \end{array}]$

Step 1.6.2

Multiply $2$ by $1$ .

$[\begin{array}{c} 5 & 20 \\ 10 & 15 \end{array}] + [\begin{array}{c} 0 & 24 \\ - 6 & 18 \end{array}] + [\begin{array}{c} 0 & 2 \\ 2 \cdot - 1 & 2 \cdot 2 \end{array}]$

Step 1.6.3

Multiply $2$ by $- 1$ .

$[\begin{array}{c} 5 & 20 \\ 10 & 15 \end{array}] + [\begin{array}{c} 0 & 24 \\ - 6 & 18 \end{array}] + [\begin{array}{c} 0 & 2 \\ - 2 & 2 \cdot 2 \end{array}]$

Step 1.6.4

Multiply $2$ by $2$ .

$[\begin{array}{c} 5 & 20 \\ 10 & 15 \end{array}] + [\begin{array}{c} 0 & 24 \\ - 6 & 18 \end{array}] + [\begin{array}{c} 0 & 2 \\ - 2 & 4 \end{array}]$

Step 2

Add the corresponding elements.

$[\begin{array}{c} 5 + 0 & 20 + 24 \\ 10 - 6 & 15 + 18 \end{array}] + [\begin{array}{c} 0 & 2 \\ - 2 & 4 \end{array}]$

Step 3

Simplify each element.

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Step 3.1

Add $5$ and $0$ .

$[\begin{array}{c} 5 & 20 + 24 \\ 10 - 6 & 15 + 18 \end{array}] + [\begin{array}{c} 0 & 2 \\ - 2 & 4 \end{array}]$

Step 3.2

Add $20$ and $24$ .

$[\begin{array}{c} 5 & 44 \\ 10 - 6 & 15 + 18 \end{array}] + [\begin{array}{c} 0 & 2 \\ - 2 & 4 \end{array}]$

Step 3.3

Subtract $6$ from $10$ .

$[\begin{array}{c} 5 & 44 \\ 4 & 15 + 18 \end{array}] + [\begin{array}{c} 0 & 2 \\ - 2 & 4 \end{array}]$

Step 3.4

Add $15$ and $18$ .

$[\begin{array}{c} 5 & 44 \\ 4 & 33 \end{array}] + [\begin{array}{c} 0 & 2 \\ - 2 & 4 \end{array}]$

Step 4

Add the corresponding elements.

$[\begin{array}{c} 5 + 0 & 44 + 2 \\ 4 - 2 & 33 + 4 \end{array}]$

Step 5

Simplify each element.

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Step 5.1

Add $5$ and $0$ .

$[\begin{array}{c} 5 & 44 + 2 \\ 4 - 2 & 33 + 4 \end{array}]$

Step 5.2

Add $44$ and $2$ .

$[\begin{array}{c} 5 & 46 \\ 4 - 2 & 33 + 4 \end{array}]$

Step 5.3

Subtract $2$ from $4$ .

$[\begin{array}{c} 5 & 46 \\ 2 & 33 + 4 \end{array}]$

Step 5.4

Add $33$ and $4$ .

$[\begin{array}{c} 5 & 46 \\ 2 & 37 \end{array}]$

Step 6

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

$5 \cdot 37 - 2 \cdot 46$

Step 7

Simplify the determinant.

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Step 7.1

Simplify each term.

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Step 7.1.1

Multiply $5$ by $37$ .

$185 - 2 \cdot 46$

Step 7.1.2

Multiply $- 2$ by $46$ .

$185 - 92$

Step 7.2

Subtract $92$ from $185$ .

$93$