Algebra Examples

$[\begin{array}{c} 4 & 5 \\ 3 & 4 \end{array}]$

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 $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$4 \cdot 4 - 3 \cdot 5$

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 $4$ by $4$ .

$16 - 3 \cdot 5$

Step 2.2.1.2

Multiply $- 3$ by $5$ .

$16 - 15$

Step 2.2.2

Subtract $15$ from $16$ .

$1$

Step 3

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

Step 4

Substitute the known values into the formula for the inverse.

$\frac{1}{1} [\begin{array}{c} 4 & - 5 \\ - 3 & 4 \end{array}]$

Step 5

Divide $1$ by $1$ .

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

Step 6

Multiply $1$ by each element of the matrix.

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

Step 7

Simplify each element in the matrix.

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Step 7.1

Multiply $4$ by $1$ .

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

Step 7.2

Multiply $- 5$ by $1$ .

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

Step 7.3

Multiply $- 3$ by $1$ .

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

Step 7.4

Multiply $4$ by $1$ .

$[\begin{array}{c} 4 & - 5 \\ - 3 & 4 \end{array}]$