Solve the Matrix Equation [[1,3],[8,5]][[x],[y]]=[[1],[9]]

$[\begin{array}{c} 1 & 3 \\ 8 & 5 \end{array}] [\begin{array}{c} x \\ y \end{array}] = [\begin{array}{c} 1 \\ 9 \end{array}]$

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 $[\begin{array}{c} 1 & 3 \\ 8 & 5 \end{array}]$ .

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

$determinant [\begin{array}{c} 1 & 3 \\ 8 & 5 \end{array}] = | \begin{array}{c} 1 & 3 \\ 8 & 5 \end{array} |$

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

$(1) (5) - 8 \cdot 3$

Simplify the determinant.

Simplify each term.

Multiply $5$ by $1$ .

$5 - 8 \cdot 3$

Multiply $- 8$ by $3$ .

$5 - 24$

Subtract $24$ from $5$ .

$- 19$

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

$\frac{1}{- 19} [\begin{array}{c} 5 & - (3) \\ - (8) & 1 \end{array}]$

Simplify each element in the matrix.

Rearrange $- (3)$ .

$\frac{1}{- 19} [\begin{array}{c} 5 & - 3 \\ - (8) & 1 \end{array}]$

Rearrange $- (8)$ .

$\frac{1}{- 19} [\begin{array}{c} 5 & - 3 \\ - 8 & 1 \end{array}]$

Multiply $\frac{1}{- 19}$ by each element of the matrix.

$[\begin{array}{c} \frac{1}{- 19} \cdot 5 & \frac{1}{- 19} \cdot - 3 \\ \frac{1}{- 19} \cdot - 8 & \frac{1}{- 19} \cdot 1 \end{array}]$

Simplify each element in the matrix.

Rearrange $\frac{1}{- 19} \cdot 5$ .

$[\begin{array}{c} - \frac{5}{19} & \frac{1}{- 19} \cdot - 3 \\ \frac{1}{- 19} \cdot - 8 & \frac{1}{- 19} \cdot 1 \end{array}]$

Rearrange $\frac{1}{- 19} \cdot - 3$ .

$[\begin{array}{c} - \frac{5}{19} & \frac{3}{19} \\ \frac{1}{- 19} \cdot - 8 & \frac{1}{- 19} \cdot 1 \end{array}]$

Rearrange $\frac{1}{- 19} \cdot - 8$ .

$[\begin{array}{c} - \frac{5}{19} & \frac{3}{19} \\ \frac{8}{19} & \frac{1}{- 19} \cdot 1 \end{array}]$

Rearrange $\frac{1}{- 19} \cdot 1$ .

$[\begin{array}{c} - \frac{5}{19} & \frac{3}{19} \\ \frac{8}{19} & - \frac{1}{19} \end{array}]$

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

$[\begin{array}{c} - \frac{5}{19} & \frac{3}{19} \\ \frac{8}{19} & - \frac{1}{19} \end{array}] \cdot ([\begin{array}{c} 1 & 3 \\ 8 & 5 \end{array}] x) = [\begin{array}{c} - \frac{5}{19} & \frac{3}{19} \\ \frac{8}{19} & - \frac{1}{19} \end{array}] \cdot [\begin{array}{c} 1 \\ 9 \end{array}]$

Simplify the left side of the equation.

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

$[\begin{array}{c} - \frac{5}{19} \cdot 1 + \frac{3}{19} \cdot 8 & - \frac{5}{19} \cdot 3 + \frac{3}{19} \cdot 5 \\ \frac{8}{19} \cdot 1 - \frac{1}{19} \cdot 8 & \frac{8}{19} \cdot 3 - \frac{1}{19} \cdot 5 \end{array}] x = [\begin{array}{c} - \frac{5}{19} & \frac{3}{19} \\ \frac{8}{19} & - \frac{1}{19} \end{array}] \cdot [\begin{array}{c} 1 \\ 9 \end{array}]$

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

$[\begin{array}{c} 1 & 0 \\ 0 & 1 \end{array}] x = [\begin{array}{c} - \frac{5}{19} & \frac{3}{19} \\ \frac{8}{19} & - \frac{1}{19} \end{array}] \cdot [\begin{array}{c} 1 \\ 9 \end{array}]$

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

$x = [\begin{array}{c} - \frac{5}{19} & \frac{3}{19} \\ \frac{8}{19} & - \frac{1}{19} \end{array}] \cdot [\begin{array}{c} 1 \\ 9 \end{array}]$

Simplify the right side of the equation.

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

$x = [\begin{array}{c} - \frac{5}{19} \cdot 1 + \frac{3}{19} \cdot 9 \\ \frac{8}{19} \cdot 1 - \frac{1}{19} \cdot 9 \end{array}]$

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

$x = [\begin{array}{c} \frac{22}{19} \\ - \frac{1}{19} \end{array}]$

The matrix is in the most simplified form.

$x = [\begin{array}{c} \frac{22}{19} \\ - \frac{1}{19} \end{array}]$

Solving for the variables in $[\begin{array}{c} x \\ y \end{array}]$ , the answer is $x = \frac{22}{19}, y = - \frac{1}{19}$ .

$x = \frac{22}{19}, y = - \frac{1}{19}$

Solve the Matrix Equation [[1,3],[8,5]][[x],[y]]=[[1],[9]]

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