Linear Algebra Examples

$[\begin{array}{c} 2 & 4 \\ 8 & 1 \end{array}] + [\begin{array}{c} - 2 & 10 \\ 3 & 5 \end{array}] + [\begin{array}{c} 5 & 20 \\ 2 & - 21 \end{array}]$

Step 1

Add the corresponding elements.

$[\begin{array}{c} 2 - 2 & 4 + 10 \\ 8 + 3 & 1 + 5 \end{array}] + [\begin{array}{c} 5 & 20 \\ 2 & - 21 \end{array}]$

Step 2

Simplify each element.

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Step 2.1

Subtract $2$ from $2$ .

$[\begin{array}{c} 0 & 4 + 10 \\ 8 + 3 & 1 + 5 \end{array}] + [\begin{array}{c} 5 & 20 \\ 2 & - 21 \end{array}]$

Step 2.2

Add $4$ and $10$ .

$[\begin{array}{c} 0 & 14 \\ 8 + 3 & 1 + 5 \end{array}] + [\begin{array}{c} 5 & 20 \\ 2 & - 21 \end{array}]$

Step 2.3

Add $8$ and $3$ .

$[\begin{array}{c} 0 & 14 \\ 11 & 1 + 5 \end{array}] + [\begin{array}{c} 5 & 20 \\ 2 & - 21 \end{array}]$

Step 2.4

Add $1$ and $5$ .

$[\begin{array}{c} 0 & 14 \\ 11 & 6 \end{array}] + [\begin{array}{c} 5 & 20 \\ 2 & - 21 \end{array}]$

Step 3

Add the corresponding elements.

$[\begin{array}{c} 0 + 5 & 14 + 20 \\ 11 + 2 & 6 - 21 \end{array}]$

Step 4

Simplify each element.

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Step 4.1

Add $0$ and $5$ .

$[\begin{array}{c} 5 & 14 + 20 \\ 11 + 2 & 6 - 21 \end{array}]$

Step 4.2

Add $14$ and $20$ .

$[\begin{array}{c} 5 & 34 \\ 11 + 2 & 6 - 21 \end{array}]$

Step 4.3

Add $11$ and $2$ .

$[\begin{array}{c} 5 & 34 \\ 13 & 6 - 21 \end{array}]$

Step 4.4

Subtract $21$ from $6$ .

$[\begin{array}{c} 5 & 34 \\ 13 & - 15 \end{array}]$

Step 5

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

$5 \cdot - 15 - 13 \cdot 34$

Step 6

Simplify the determinant.

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Step 6.1

Simplify each term.

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Step 6.1.1

Multiply $5$ by $- 15$ .

$- 75 - 13 \cdot 34$

Step 6.1.2

Multiply $- 13$ by $34$ .

$- 75 - 442$

Step 6.2

Subtract $442$ from $- 75$ .

$- 517$