Solve the Matrix Equation [[9,5],[7,3]]X=[[-4],[-5]]

$[\begin{array}{c} 9 & 5 \\ 7 & 3 \end{array}] X = [\begin{array}{c} - 4 \\ - 5 \end{array}]$

Step 1

Find the inverse of $[\begin{array}{c} 9 & 5 \\ 7 & 3 \end{array}]$ .

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.

Find the determinant.

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

$9 \cdot 3 - 7 \cdot 5$

Simplify the determinant.

Simplify each term.

Multiply $9$ by $3$ .

$27 - 7 \cdot 5$

Multiply $- 7$ by $5$ .

$27 - 35$

Subtract $35$ from $27$ .

$- 8$

Substitute the known values into the formula for the inverse.

$\frac{1}{- 8} [\begin{array}{c} 3 & - 5 \\ - 7 & 9 \end{array}]$

Move the negative in front of the fraction.

$- \frac{1}{8} [\begin{array}{c} 3 & - 5 \\ - 7 & 9 \end{array}]$

Multiply $- \frac{1}{8}$ by each element of the matrix.

$[\begin{array}{c} - \frac{1}{8} \cdot 3 & - \frac{1}{8} \cdot - 5 \\ - \frac{1}{8} \cdot - 7 & - \frac{1}{8} \cdot 9 \end{array}]$

Simplify each element in the matrix.

Multiply $- \frac{1}{8} \cdot 3$ .

Multiply $3$ by $- 1$ .

$[\begin{array}{c} - 3 (\frac{1}{8}) & - \frac{1}{8} \cdot - 5 \\ - \frac{1}{8} \cdot - 7 & - \frac{1}{8} \cdot 9 \end{array}]$

Combine $- 3$ and $\frac{1}{8}$ .

$[\begin{array}{c} \frac{- 3}{8} & - \frac{1}{8} \cdot - 5 \\ - \frac{1}{8} \cdot - 7 & - \frac{1}{8} \cdot 9 \end{array}]$

Move the negative in front of the fraction.

$[\begin{array}{c} - \frac{3}{8} & - \frac{1}{8} \cdot - 5 \\ - \frac{1}{8} \cdot - 7 & - \frac{1}{8} \cdot 9 \end{array}]$

Multiply $- \frac{1}{8} \cdot - 5$ .

Multiply $- 5$ by $- 1$ .

$[\begin{array}{c} - \frac{3}{8} & 5 (\frac{1}{8}) \\ - \frac{1}{8} \cdot - 7 & - \frac{1}{8} \cdot 9 \end{array}]$

Combine $5$ and $\frac{1}{8}$ .

$[\begin{array}{c} - \frac{3}{8} & \frac{5}{8} \\ - \frac{1}{8} \cdot - 7 & - \frac{1}{8} \cdot 9 \end{array}]$

Multiply $- \frac{1}{8} \cdot - 7$ .

Multiply $- 7$ by $- 1$ .

$[\begin{array}{c} - \frac{3}{8} & \frac{5}{8} \\ 7 (\frac{1}{8}) & - \frac{1}{8} \cdot 9 \end{array}]$

Combine $7$ and $\frac{1}{8}$ .

$[\begin{array}{c} - \frac{3}{8} & \frac{5}{8} \\ \frac{7}{8} & - \frac{1}{8} \cdot 9 \end{array}]$

Multiply $- \frac{1}{8} \cdot 9$ .

Multiply $9$ by $- 1$ .

$[\begin{array}{c} - \frac{3}{8} & \frac{5}{8} \\ \frac{7}{8} & - 9 (\frac{1}{8}) \end{array}]$

Combine $- 9$ and $\frac{1}{8}$ .

$[\begin{array}{c} - \frac{3}{8} & \frac{5}{8} \\ \frac{7}{8} & \frac{- 9}{8} \end{array}]$

Move the negative in front of the fraction.

$[\begin{array}{c} - \frac{3}{8} & \frac{5}{8} \\ \frac{7}{8} & - \frac{9}{8} \end{array}]$

Step 2

Multiply both sides by the inverse of $[\begin{array}{c} 9 & 5 \\ 7 & 3 \end{array}]$ .

$[\begin{array}{c} - \frac{3}{8} & \frac{5}{8} \\ \frac{7}{8} & - \frac{9}{8} \end{array}] [\begin{array}{c} 9 & 5 \\ 7 & 3 \end{array}] X = [\begin{array}{c} - \frac{3}{8} & \frac{5}{8} \\ \frac{7}{8} & - \frac{9}{8} \end{array}] [\begin{array}{c} - 4 \\ - 5 \end{array}]$

Step 3

Simplify the equation.

Multiply $[\begin{array}{c} - \frac{3}{8} & \frac{5}{8} \\ \frac{7}{8} & - \frac{9}{8} \end{array}] [\begin{array}{c} 9 & 5 \\ 7 & 3 \end{array}]$ .

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

$[\begin{array}{c} - \frac{3}{8} \cdot 9 + \frac{5}{8} \cdot 7 & - \frac{3}{8} \cdot 5 + \frac{5}{8} \cdot 3 \\ \frac{7}{8} \cdot 9 - \frac{9}{8} \cdot 7 & \frac{7}{8} \cdot 5 - \frac{9}{8} \cdot 3 \end{array}] X = [\begin{array}{c} - \frac{3}{8} & \frac{5}{8} \\ \frac{7}{8} & - \frac{9}{8} \end{array}] [\begin{array}{c} - 4 \\ - 5 \end{array}]$

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

$[\begin{array}{c} 1 & 0 \\ 0 & 1 \end{array}] X = [\begin{array}{c} - \frac{3}{8} & \frac{5}{8} \\ \frac{7}{8} & - \frac{9}{8} \end{array}] [\begin{array}{c} - 4 \\ - 5 \end{array}]$

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

$X = [\begin{array}{c} - \frac{3}{8} & \frac{5}{8} \\ \frac{7}{8} & - \frac{9}{8} \end{array}] [\begin{array}{c} - 4 \\ - 5 \end{array}]$

Multiply $[\begin{array}{c} - \frac{3}{8} & \frac{5}{8} \\ \frac{7}{8} & - \frac{9}{8} \end{array}] [\begin{array}{c} - 4 \\ - 5 \end{array}]$ .

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

$X = [\begin{array}{c} - \frac{3}{8} \cdot - 4 + \frac{5}{8} \cdot - 5 \\ \frac{7}{8} \cdot - 4 - \frac{9}{8} \cdot - 5 \end{array}]$

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

Multiply $- 7$ by $4$ .

$X = [\begin{array}{c} - \frac{13}{8} \\ \frac{- 28 + 45}{8} \end{array}]$

Add $- 28$ and $45$ .

$X = [\begin{array}{c} - \frac{13}{8} \\ \frac{17}{8} \end{array}]$

Solve the Matrix Equation [[9,5],[7,3]]X=[[-4],[-5]]

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