Linear Algebra Examples

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

Step 1

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 2

Find the determinant.

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Step 2.1

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

$2 \cdot 2 + 0 \cdot 0$

Step 2.2

Simplify the determinant.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Multiply $2$ by $2$ .

$4 + 0 \cdot 0$

Step 2.2.1.2

Multiply $0$ by $0$ .

$4 + 0$

Step 2.2.2

Add $4$ and $0$ .

$4$

Step 3

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

Step 4

Substitute the known values into the formula for the inverse.

$\frac{1}{4} [\begin{array}{c} 2 & 0 \\ 0 & 2 \end{array}]$

Step 5

Multiply $\frac{1}{4}$ by each element of the matrix.

$[\begin{array}{c} \frac{1}{4} \cdot 2 & \frac{1}{4} \cdot 0 \\ \frac{1}{4} \cdot 0 & \frac{1}{4} \cdot 2 \end{array}]$

Step 6

Simplify each element in the matrix.

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Step 6.1

Cancel the common factor of $2$ .

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Step 6.1.1

Factor $2$ out of $4$ .

$[\begin{array}{c} \frac{1}{2 (2)} \cdot 2 & \frac{1}{4} \cdot 0 \\ \frac{1}{4} \cdot 0 & \frac{1}{4} \cdot 2 \end{array}]$

Step 6.1.2

Cancel the common factor.

$[\begin{array}{c} \frac{1}{2 \cdot 2} \cdot 2 & \frac{1}{4} \cdot 0 \\ \frac{1}{4} \cdot 0 & \frac{1}{4} \cdot 2 \end{array}]$

Step 6.1.3

Rewrite the expression.

$[\begin{array}{c} \frac{1}{2} & \frac{1}{4} \cdot 0 \\ \frac{1}{4} \cdot 0 & \frac{1}{4} \cdot 2 \end{array}]$

Step 6.2

Multiply $\frac{1}{4}$ by $0$ .

$[\begin{array}{c} \frac{1}{2} & 0 \\ \frac{1}{4} \cdot 0 & \frac{1}{4} \cdot 2 \end{array}]$

Step 6.3

Multiply $\frac{1}{4}$ by $0$ .

$[\begin{array}{c} \frac{1}{2} & 0 \\ 0 & \frac{1}{4} \cdot 2 \end{array}]$

Step 6.4

Cancel the common factor of $2$ .

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Step 6.4.1

Factor $2$ out of $4$ .

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

Step 6.4.2

Cancel the common factor.

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

Step 6.4.3

Rewrite the expression.

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