Algebra Examples

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

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

Step 1

The inverse of a

2 \times 2

$2 \times 2$ matrix can be found using the formula

\frac{1}{a d - b c} [\begin{array}{c} d & - b \\ - c & a \end{array}]

$\frac{1}{a d - b c} [\begin{array}{c} d & - b \\ - c & a \end{array}]$ where

a d - b c

$a d - b c$ is the determinant.

Step 2

Find the determinant.

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Step 2.1

The determinant of a

2 \times 2

$2 \times 2$ matrix can be found using the formula

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

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

0 \cdot 0 - 1 \cdot 1

$0 \cdot 0 - 1 \cdot 1$

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

0

$0$ by

0

$0$ .

0 - 1 \cdot 1

$0 - 1 \cdot 1$

Step 2.2.1.2

Multiply

- 1

$- 1$ by

1

$1$ .

0 - 1

$0 - 1$

0 - 1

$0 - 1$

Step 2.2.2

Subtract

1

$1$ from

0

$0$ .

- 1

$- 1$

- 1

$- 1$

- 1

$- 1$

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} 0 & - 1 \\ - 1 & 0 \end{array}]

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

Step 5

Divide

1

$1$ by

- 1

$- 1$ .

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

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

Step 6

Multiply

- 1

$- 1$ by each element of the matrix.

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

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

Step 7

Simplify each element in the matrix.

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Step 7.1

Multiply

- 1

$- 1$ by

0

$0$ .

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

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

Step 7.2

Multiply

- 1

$- 1$ by

- 1

$- 1$ .

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

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

Step 7.3

Multiply

- 1

$- 1$ by

- 1

$- 1$ .

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

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

Step 7.4

Multiply

- 1

$- 1$ by

0

$0$ .

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

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

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

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

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