Solve the Matrix Equation [[9,-5],[11,-5]]X=[[22],[28]]

$[\begin{array}{c} 9 & - 5 \\ 11 & - 5 \end{array}] X = [\begin{array}{c} 22 \\ 28 \end{array}]$

Step 1

Find the inverse matrix.

The inverse of a $2 \times 2$ matrix can be found using the formula $\frac{1}{| A |} [\begin{array}{c} d & - b \\ - c & a \end{array}]$ where $| A |$ is the determinant of $A$ .

If $A = [\begin{array}{c} a & b \\ c & d \end{array}]$ then $A^{- 1} = \frac{1}{| A |} [\begin{array}{c} d & - b \\ - c & a \end{array}]$

Find the determinant of $[\begin{array}{c} 9 & - 5 \\ 11 & - 5 \end{array}]$ .

These are both valid notations for the determinant of a matrix.

$determinant [\begin{array}{c} 9 & - 5 \\ 11 & - 5 \end{array}] = | \begin{array}{c} 9 & - 5 \\ 11 & - 5 \end{array} |$

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

$(9) (- 5) - 11 \cdot - 5$

Simplify the determinant.

Simplify each term.

Multiply $9$ by $- 5$ .

$- 45 - 11 \cdot - 5$

Multiply $- 11$ by $- 5$ .

$- 45 + 55$

Add $- 45$ and $55$ .

$10$

Substitute the known values into the formula for the inverse of a matrix.

$\frac{1}{10} [\begin{array}{c} - 5 & - (- 5) \\ - (11) & 9 \end{array}]$

Simplify each element in the matrix.

Rearrange $- (- 5)$ .

$\frac{1}{10} [\begin{array}{c} - 5 & 5 \\ - (11) & 9 \end{array}]$

Rearrange $- (11)$ .

$\frac{1}{10} [\begin{array}{c} - 5 & 5 \\ - 11 & 9 \end{array}]$

Multiply $\frac{1}{10}$ by each element of the matrix.

$[\begin{array}{c} \frac{1}{10} \cdot - 5 & \frac{1}{10} \cdot 5 \\ \frac{1}{10} \cdot - 11 & \frac{1}{10} \cdot 9 \end{array}]$

Simplify each element in the matrix.

Rearrange $\frac{1}{10} \cdot - 5$ .

$[\begin{array}{c} - \frac{1}{2} & \frac{1}{10} \cdot 5 \\ \frac{1}{10} \cdot - 11 & \frac{1}{10} \cdot 9 \end{array}]$

Rearrange $\frac{1}{10} \cdot 5$ .

$[\begin{array}{c} - \frac{1}{2} & \frac{1}{2} \\ \frac{1}{10} \cdot - 11 & \frac{1}{10} \cdot 9 \end{array}]$

Rearrange $\frac{1}{10} \cdot - 11$ .

$[\begin{array}{c} - \frac{1}{2} & \frac{1}{2} \\ - \frac{11}{10} & \frac{1}{10} \cdot 9 \end{array}]$

Rearrange $\frac{1}{10} \cdot 9$ .

$[\begin{array}{c} - \frac{1}{2} & \frac{1}{2} \\ - \frac{11}{10} & \frac{9}{10} \end{array}]$

Step 2

Assuming that $X$ is the matrix to solve for, multiply the inverse matrix by both sides of the equation.

$[\begin{array}{c} - \frac{1}{2} & \frac{1}{2} \\ - \frac{11}{10} & \frac{9}{10} \end{array}] \cdot ([\begin{array}{c} 9 & - 5 \\ 11 & - 5 \end{array}] x) = [\begin{array}{c} - \frac{1}{2} & \frac{1}{2} \\ - \frac{11}{10} & \frac{9}{10} \end{array}] \cdot [\begin{array}{c} 22 \\ 28 \end{array}]$

Step 3

Simplify the left side of the equation.

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

$[\begin{array}{c} - \frac{1}{2} \cdot 9 + \frac{1}{2} \cdot 11 & - \frac{1}{2} \cdot - 5 + \frac{1}{2} \cdot - 5 \\ - \frac{11}{10} \cdot 9 + \frac{9}{10} \cdot 11 & - \frac{11}{10} \cdot - 5 + \frac{9}{10} \cdot - 5 \end{array}] x = [\begin{array}{c} - \frac{1}{2} & \frac{1}{2} \\ - \frac{11}{10} & \frac{9}{10} \end{array}] \cdot [\begin{array}{c} 22 \\ 28 \end{array}]$

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

$[\begin{array}{c} 1 & 0 \\ 0 & 1 \end{array}] x = [\begin{array}{c} - \frac{1}{2} & \frac{1}{2} \\ - \frac{11}{10} & \frac{9}{10} \end{array}] \cdot [\begin{array}{c} 22 \\ 28 \end{array}]$

Multiplying the identity matrix by any matrix $x$ is matrix $x$ .

$x = [\begin{array}{c} - \frac{1}{2} & \frac{1}{2} \\ - \frac{11}{10} & \frac{9}{10} \end{array}] \cdot [\begin{array}{c} 22 \\ 28 \end{array}]$

Step 4

Simplify the right side of the equation.

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

$x = [\begin{array}{c} - \frac{1}{2} \cdot 22 + \frac{1}{2} \cdot 28 \\ - \frac{11}{10} \cdot 22 + \frac{9}{10} \cdot 28 \end{array}]$

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

$x = [\begin{array}{c} 3 \\ 1 \end{array}]$

The matrix is in the most simplified form.

$x = [\begin{array}{c} 3 \\ 1 \end{array}]$

Solve the Matrix Equation [[9,-5],[11,-5]]X=[[22],[28]]

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