Linear Algebra Examples

$[\begin{array}{c} 9 & 4 \\ 2 & 1 \end{array}] X = [\begin{array}{c} 0 & - 1 \\ 3 & 2 \end{array}]$

Step 1

Find the inverse of $[\begin{array}{c} 9 & 4 \\ 2 & 1 \end{array}]$ .

Tap for more steps...

Step 1.1

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 1.2

Find the determinant.

Tap for more steps...

Step 1.2.1

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

$9 \cdot 1 - 2 \cdot 4$

Step 1.2.2

Simplify the determinant.

Tap for more steps...

Step 1.2.2.1

Simplify each term.

Tap for more steps...

Step 1.2.2.1.1

Multiply $9$ by $1$ .

$9 - 2 \cdot 4$

Step 1.2.2.1.2

Multiply $- 2$ by $4$ .

$9 - 8$

Step 1.2.2.2

Subtract $8$ from $9$ .

$1$

Step 1.3

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

Step 1.4

Substitute the known values into the formula for the inverse.

$\frac{1}{1} [\begin{array}{c} 1 & - 4 \\ - 2 & 9 \end{array}]$

Step 1.5

Divide $1$ by $1$ .

$1 [\begin{array}{c} 1 & - 4 \\ - 2 & 9 \end{array}]$

Step 1.6

Multiply $1$ by each element of the matrix.

$[\begin{array}{c} 1 \cdot 1 & 1 \cdot - 4 \\ 1 \cdot - 2 & 1 \cdot 9 \end{array}]$

Step 1.7

Simplify each element in the matrix.

Tap for more steps...

Step 1.7.1

Multiply $1$ by $1$ .

$[\begin{array}{c} 1 & 1 \cdot - 4 \\ 1 \cdot - 2 & 1 \cdot 9 \end{array}]$

Step 1.7.2

Multiply $- 4$ by $1$ .

$[\begin{array}{c} 1 & - 4 \\ 1 \cdot - 2 & 1 \cdot 9 \end{array}]$

Step 1.7.3

Multiply $- 2$ by $1$ .

$[\begin{array}{c} 1 & - 4 \\ - 2 & 1 \cdot 9 \end{array}]$

Step 1.7.4

Multiply $9$ by $1$ .

$[\begin{array}{c} 1 & - 4 \\ - 2 & 9 \end{array}]$

Step 2

Multiply both sides by the inverse of $[\begin{array}{c} 9 & 4 \\ 2 & 1 \end{array}]$ .

$[\begin{array}{c} 1 & - 4 \\ - 2 & 9 \end{array}] [\begin{array}{c} 9 & 4 \\ 2 & 1 \end{array}] X = [\begin{array}{c} 1 & - 4 \\ - 2 & 9 \end{array}] [\begin{array}{c} 0 & - 1 \\ 3 & 2 \end{array}]$

Step 3

Simplify the equation.

Tap for more steps...

Step 3.1

Multiply $[\begin{array}{c} 1 & - 4 \\ - 2 & 9 \end{array}] [\begin{array}{c} 9 & 4 \\ 2 & 1 \end{array}]$ .

Tap for more steps...

Step 3.1.1

Two matrices can be multiplied if and only if the number of columns in the first matrix is equal to the number of rows in the second matrix. In this case, the first matrix is $2 \times 2$ and the second matrix is $2 \times 2$ .

Step 3.1.2

Multiply each row in the first matrix by each column in the second matrix.

$[\begin{array}{c} 1 \cdot 9 - 4 \cdot 2 & 1 \cdot 4 - 4 \cdot 1 \\ - 2 \cdot 9 + 9 \cdot 2 & - 2 \cdot 4 + 9 \cdot 1 \end{array}] X = [\begin{array}{c} 1 & - 4 \\ - 2 & 9 \end{array}] [\begin{array}{c} 0 & - 1 \\ 3 & 2 \end{array}]$

Step 3.1.3

Simplify each element of the matrix by multiplying out all the expressions.

$[\begin{array}{c} 1 & 0 \\ 0 & 1 \end{array}] X = [\begin{array}{c} 1 & - 4 \\ - 2 & 9 \end{array}] [\begin{array}{c} 0 & - 1 \\ 3 & 2 \end{array}]$

Step 3.2

Multiplying the identity matrix by any matrix $A$ is the matrix $A$ itself.

$X = [\begin{array}{c} 1 & - 4 \\ - 2 & 9 \end{array}] [\begin{array}{c} 0 & - 1 \\ 3 & 2 \end{array}]$

Step 3.3

Multiply $[\begin{array}{c} 1 & - 4 \\ - 2 & 9 \end{array}] [\begin{array}{c} 0 & - 1 \\ 3 & 2 \end{array}]$ .

Tap for more steps...

Step 3.3.1

Step 3.3.2

Multiply each row in the first matrix by each column in the second matrix.

$X = [\begin{array}{c} 1 \cdot 0 - 4 \cdot 3 & 1 \cdot - 1 - 4 \cdot 2 \\ - 2 \cdot 0 + 9 \cdot 3 & - 2 \cdot - 1 + 9 \cdot 2 \end{array}]$

Step 3.3.3

Simplify each element of the matrix by multiplying out all the expressions.

$X = [\begin{array}{c} - 12 & - 9 \\ 27 & 20 \end{array}]$