Linear Algebra Examples

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

Step 1

Add the corresponding elements.

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

Step 2

Simplify each element.

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Step 2.1

Subtract $1$ from $1$ .

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

Step 2.2

Subtract $1$ from $0$ .

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

Step 2.3

Add $0$ and $2$ .

$[\begin{array}{c} 0 & - 1 \\ 2 & 1 - 2 \end{array}]$

Step 2.4

Subtract $2$ from $1$ .

$[\begin{array}{c} 0 & - 1 \\ 2 & - 1 \end{array}]$

Step 3

The inverse of a $2 \times 2$ matrix can be found using the formula $\frac{1}{a d - b c} [\begin{array}{c} d & - b \\ - c & a \end{array}]$ where $a d - b c$ is the determinant.

Step 4

Find the determinant.

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Step 4.1

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

$0 \cdot - 1 - 2 \cdot - 1$

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 $0$ by $- 1$ .

$0 - 2 \cdot - 1$

Step 4.2.1.2

Multiply $- 2$ by $- 1$ .

$0 + 2$

Step 4.2.2

Add $0$ and $2$ .

$2$

Step 5

Since the determinant is non-zero, the inverse exists.

Step 6

Substitute the known values into the formula for the inverse.

$\frac{1}{2} [\begin{array}{c} - 1 & 1 \\ - 2 & 0 \end{array}]$

Step 7

Multiply $\frac{1}{2}$ by each element of the matrix.

$[\begin{array}{c} \frac{1}{2} \cdot - 1 & \frac{1}{2} \cdot 1 \\ \frac{1}{2} \cdot - 2 & \frac{1}{2} \cdot 0 \end{array}]$

Step 8

Simplify each element in the matrix.

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Step 8.1

Combine $\frac{1}{2}$ and $- 1$ .

$[\begin{array}{c} \frac{- 1}{2} & \frac{1}{2} \cdot 1 \\ \frac{1}{2} \cdot - 2 & \frac{1}{2} \cdot 0 \end{array}]$

Step 8.2

Move the negative in front of the fraction.

$[\begin{array}{c} - \frac{1}{2} & \frac{1}{2} \cdot 1 \\ \frac{1}{2} \cdot - 2 & \frac{1}{2} \cdot 0 \end{array}]$

Step 8.3

Multiply $\frac{1}{2}$ by $1$ .

$[\begin{array}{c} - \frac{1}{2} & \frac{1}{2} \\ \frac{1}{2} \cdot - 2 & \frac{1}{2} \cdot 0 \end{array}]$

Step 8.4

Cancel the common factor of $2$ .

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Step 8.4.1

Factor $2$ out of $- 2$ .

$[\begin{array}{c} - \frac{1}{2} & \frac{1}{2} \\ \frac{1}{2} \cdot (2 (- 1)) & \frac{1}{2} \cdot 0 \end{array}]$

Step 8.4.2

Cancel the common factor.

$[\begin{array}{c} - \frac{1}{2} & \frac{1}{2} \\ \frac{1}{2} \cdot (2 \cdot - 1) & \frac{1}{2} \cdot 0 \end{array}]$

Step 8.4.3

Rewrite the expression.

$[\begin{array}{c} - \frac{1}{2} & \frac{1}{2} \\ - 1 & \frac{1}{2} \cdot 0 \end{array}]$

Step 8.5

Multiply $\frac{1}{2}$ by $0$ .

$[\begin{array}{c} - \frac{1}{2} & \frac{1}{2} \\ - 1 & 0 \end{array}]$

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