Solve the Matrix Equation [[5,4],[-3,2]]X=[[10],[-16]]

$[\begin{array}{c} 5 & 4 \\ - 3 & 2 \end{array}] X = [\begin{array}{c} 10 \\ - 16 \end{array}]$

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 $[\begin{array}{c} 5 & 4 \\ - 3 & 2 \end{array}]$ .

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

$determinant [\begin{array}{c} 5 & 4 \\ - 3 & 2 \end{array}] = | \begin{array}{c} 5 & 4 \\ - 3 & 2 \end{array} |$

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

$(5) (2) + 3 \cdot 4$

Simplify the determinant.

Simplify each term.

Multiply $5$ by $2$ .

$10 + 3 \cdot 4$

Multiply $3$ by $4$ .

$10 + 12$

Add $10$ and $12$ .

$22$

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

$\frac{1}{22} [\begin{array}{c} 2 & - (4) \\ - (- 3) & 5 \end{array}]$

Simplify each element in the matrix.

Rearrange $- (4)$ .

$\frac{1}{22} [\begin{array}{c} 2 & - 4 \\ - (- 3) & 5 \end{array}]$

Rearrange $- (- 3)$ .

$\frac{1}{22} [\begin{array}{c} 2 & - 4 \\ 3 & 5 \end{array}]$

Multiply $\frac{1}{22}$ by each element of the matrix.

$[\begin{array}{c} \frac{1}{22} \cdot 2 & \frac{1}{22} \cdot - 4 \\ \frac{1}{22} \cdot 3 & \frac{1}{22} \cdot 5 \end{array}]$

Simplify each element in the matrix.

Rearrange $\frac{1}{22} \cdot 2$ .

$[\begin{array}{c} \frac{1}{11} & \frac{1}{22} \cdot - 4 \\ \frac{1}{22} \cdot 3 & \frac{1}{22} \cdot 5 \end{array}]$

Rearrange $\frac{1}{22} \cdot - 4$ .

$[\begin{array}{c} \frac{1}{11} & - \frac{2}{11} \\ \frac{1}{22} \cdot 3 & \frac{1}{22} \cdot 5 \end{array}]$

Rearrange $\frac{1}{22} \cdot 3$ .

$[\begin{array}{c} \frac{1}{11} & - \frac{2}{11} \\ \frac{3}{22} & \frac{1}{22} \cdot 5 \end{array}]$

Rearrange $\frac{1}{22} \cdot 5$ .

$[\begin{array}{c} \frac{1}{11} & - \frac{2}{11} \\ \frac{3}{22} & \frac{5}{22} \end{array}]$

Step 2

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

$[\begin{array}{c} \frac{1}{11} & - \frac{2}{11} \\ \frac{3}{22} & \frac{5}{22} \end{array}] \cdot ([\begin{array}{c} 5 & 4 \\ - 3 & 2 \end{array}] x) = [\begin{array}{c} \frac{1}{11} & - \frac{2}{11} \\ \frac{3}{22} & \frac{5}{22} \end{array}] \cdot [\begin{array}{c} 10 \\ - 16 \end{array}]$

Step 3

Simplify the left side of the equation.

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

$[\begin{array}{c} \frac{1}{11} \cdot 5 - \frac{2}{11} \cdot - 3 & \frac{1}{11} \cdot 4 - \frac{2}{11} \cdot 2 \\ \frac{3}{22} \cdot 5 + \frac{5}{22} \cdot - 3 & \frac{3}{22} \cdot 4 + \frac{5}{22} \cdot 2 \end{array}] x = [\begin{array}{c} \frac{1}{11} & - \frac{2}{11} \\ \frac{3}{22} & \frac{5}{22} \end{array}] \cdot [\begin{array}{c} 10 \\ - 16 \end{array}]$

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

$[\begin{array}{c} 1 & 0 \\ 0 & 1 \end{array}] x = [\begin{array}{c} \frac{1}{11} & - \frac{2}{11} \\ \frac{3}{22} & \frac{5}{22} \end{array}] \cdot [\begin{array}{c} 10 \\ - 16 \end{array}]$

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

$x = [\begin{array}{c} \frac{1}{11} & - \frac{2}{11} \\ \frac{3}{22} & \frac{5}{22} \end{array}] \cdot [\begin{array}{c} 10 \\ - 16 \end{array}]$

Step 4

Simplify the right side of the equation.

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

$x = [\begin{array}{c} \frac{1}{11} \cdot 10 - \frac{2}{11} \cdot - 16 \\ \frac{3}{22} \cdot 10 + \frac{5}{22} \cdot - 16 \end{array}]$

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

$x = [\begin{array}{c} \frac{42}{11} \\ - \frac{25}{11} \end{array}]$

The matrix is in the most simplified form.

$x = [\begin{array}{c} \frac{42}{11} \\ - \frac{25}{11} \end{array}]$

Solve the Matrix Equation [[5,4],[-3,2]]X=[[10],[-16]]

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