Algebra Examples

[\begin{matrix} 4 & - 3 0 & 1 \end{matrix}] - [\begin{matrix} 4 & - 1 - 12 & - 2 \end{matrix}]

$[\begin{array}{c} 4 & - 3 \\ 0 & 1 \end{array}] - [\begin{array}{c} 4 & - 1 \\ - 12 & - 2 \end{array}]$

Step 1

Subtract the corresponding elements.

[\begin{matrix} 4 - 4 & - 3 + 1 0 + 12 & 1 + 2 \end{matrix}]

$[\begin{array}{c} 4 - 4 & - 3 + 1 \\ 0 + 12 & 1 + 2 \end{array}]$

Step 2

Simplify each element.

Tap for more steps...

Step 2.1

Subtract

4

$4$ from

4

$4$ .

[\begin{matrix} 0 & - 3 + 1 0 + 12 & 1 + 2 \end{matrix}]

$[\begin{array}{c} 0 & - 3 + 1 \\ 0 + 12 & 1 + 2 \end{array}]$

Step 2.2

Add

- 3

$- 3$ and

1

$1$ .

[\begin{matrix} 0 & - 2 0 + 12 & 1 + 2 \end{matrix}]

$[\begin{array}{c} 0 & - 2 \\ 0 + 12 & 1 + 2 \end{array}]$

Step 2.3

Add

0

$0$ and

12

$12$ .

[\begin{matrix} 0 & - 2 12 & 1 + 2 \end{matrix}]

$[\begin{array}{c} 0 & - 2 \\ 12 & 1 + 2 \end{array}]$

Step 2.4

Add

1

$1$ and

2

$2$ .

[\begin{matrix} 0 & - 2 12 & 3 \end{matrix}]

$[\begin{array}{c} 0 & - 2 \\ 12 & 3 \end{array}]$

[\begin{matrix} 0 & - 2 12 & 3 \end{matrix}]

$[\begin{array}{c} 0 & - 2 \\ 12 & 3 \end{array}]$

Step 3

The inverse of a

2 \times 2

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

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

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

a d - b c

$a d - b c$ is the determinant.

Step 4

Find the determinant.

Tap for more steps...

Step 4.1

The determinant of a

2 \times 2

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

∣ ∣ ∣ \begin{matrix} a & b c & d \end{matrix} ∣ ∣ ∣ = a d - c b

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

0 \cdot 3 - 12 \cdot - 2

$0 \cdot 3 - 12 \cdot - 2$

Step 4.2

Simplify the determinant.

Tap for more steps...

Step 4.2.1

Simplify each term.

Tap for more steps...

Step 4.2.1.1

Multiply

0

$0$ by

3

$3$ .

0 - 12 \cdot - 2

$0 - 12 \cdot - 2$

Step 4.2.1.2

Multiply

- 12

$- 12$ by

- 2

$- 2$ .

0 + 24

$0 + 24$

0 + 24

$0 + 24$

Step 4.2.2

Add

0

$0$ and

24

$24$ .

24

$24$

24

$24$

24

$24$

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}{24} [\begin{matrix} 3 & 2 - 12 & 0 \end{matrix}]

$\frac{1}{24} [\begin{array}{c} 3 & 2 \\ - 12 & 0 \end{array}]$

Step 7

Multiply

\frac{1}{24}

$\frac{1}{24}$ by each element of the matrix.

[\begin{matrix} \frac{1}{24} \cdot 3 & \frac{1}{24} \cdot 2 \frac{1}{24} \cdot - 12 & \frac{1}{24} \cdot 0 \end{matrix}]

$[\begin{array}{c} \frac{1}{24} \cdot 3 & \frac{1}{24} \cdot 2 \\ \frac{1}{24} \cdot - 12 & \frac{1}{24} \cdot 0 \end{array}]$

Step 8

Simplify each element in the matrix.

Tap for more steps...

Step 8.1

Cancel the common factor of

3

$3$ .

Tap for more steps...

Step 8.1.1

Factor

3

$3$ out of

24

$24$ .

⎡ ⎣ \begin{matrix} \frac{1}{3 (8)} \cdot 3 & \frac{1}{24} \cdot 2 \frac{1}{24} \cdot - 12 & \frac{1}{24} \cdot 0 \end{matrix} ⎤ ⎦

$[\begin{array}{c} \frac{1}{3 (8)} \cdot 3 & \frac{1}{24} \cdot 2 \\ \frac{1}{24} \cdot - 12 & \frac{1}{24} \cdot 0 \end{array}]$

Step 8.1.2

Cancel the common factor.

$[\begin{array}{c} \frac{1}{3 \cdot 8} \cdot 3 & \frac{1}{24} \cdot 2 \\ \frac{1}{24} \cdot - 12 & \frac{1}{24} \cdot 0 \end{array}]$

Step 8.1.3

Rewrite the expression.

$[\begin{array}{c} \frac{1}{8} & \frac{1}{24} \cdot 2 \\ \frac{1}{24} \cdot - 12 & \frac{1}{24} \cdot 0 \end{array}]$

Step 8.2

Cancel the common factor of

$2$ .

Tap for more steps...

Step 8.2.1

Factor

$2$ out of

$24$ .

$[\begin{array}{c} \frac{1}{8} & \frac{1}{2 (12)} \cdot 2 \\ \frac{1}{24} \cdot - 12 & \frac{1}{24} \cdot 0 \end{array}]$

Step 8.2.2

Cancel the common factor.

$[\begin{array}{c} \frac{1}{8} & \frac{1}{2 \cdot 12} \cdot 2 \\ \frac{1}{24} \cdot - 12 & \frac{1}{24} \cdot 0 \end{array}]$

Step 8.2.3

Rewrite the expression.

$[\begin{array}{c} \frac{1}{8} & \frac{1}{12} \\ \frac{1}{24} \cdot - 12 & \frac{1}{24} \cdot 0 \end{array}]$

Step 8.3

Cancel the common factor of

$12$ .

Tap for more steps...

Step 8.3.1

Factor

$12$ out of

$24$ .

$[\begin{array}{c} \frac{1}{8} & \frac{1}{12} \\ \frac{1}{12 (2)} \cdot - 12 & \frac{1}{24} \cdot 0 \end{array}]$

Step 8.3.2

Factor

$12$ out of

$- 12$ .

$[\begin{array}{c} \frac{1}{8} & \frac{1}{12} \\ \frac{1}{12 \cdot 2} \cdot (12 \cdot - 1) & \frac{1}{24} \cdot 0 \end{array}]$

Step 8.3.3

Cancel the common factor.

$[\begin{array}{c} \frac{1}{8} & \frac{1}{12} \\ \frac{1}{12 \cdot 2} \cdot (12 \cdot - 1) & \frac{1}{24} \cdot 0 \end{array}]$

Step 8.3.4

Rewrite the expression.

$[\begin{array}{c} \frac{1}{8} & \frac{1}{12} \\ \frac{1}{2} \cdot - 1 & \frac{1}{24} \cdot 0 \end{array}]$

Step 8.4

Combine

$\frac{1}{2}$ and

$- 1$ .

$[\begin{array}{c} \frac{1}{8} & \frac{1}{12} \\ \frac{- 1}{2} & \frac{1}{24} \cdot 0 \end{array}]$

Step 8.5

Move the negative in front of the fraction.

$[\begin{array}{c} \frac{1}{8} & \frac{1}{12} \\ - \frac{1}{2} & \frac{1}{24} \cdot 0 \end{array}]$

Step 8.6

Multiply

$\frac{1}{24}$ by

$0$ .

$[\begin{array}{c} \frac{1}{8} & \frac{1}{12} \\ - \frac{1}{2} & 0 \end{array}]$

Enter YOUR Problem