Solve the Matrix Equation [[9,5],[7,-4]]x=[[2],[1]]

$[\begin{array}{c} 9 & 5 \\ 7 & - 4 \end{array}] x = [\begin{array}{c} 2 \\ 1 \end{array}]$

Step 1

Find the inverse of $[\begin{array}{c} 9 & 5 \\ 7 & - 4 \end{array}]$ .

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 $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$9 \cdot - 4 - 7 \cdot 5$

Simplify the determinant.

Simplify each term.

Multiply $9$ by $- 4$ .

$- 36 - 7 \cdot 5$

Multiply $- 7$ by $5$ .

$- 36 - 35$

Subtract $35$ from $- 36$ .

$- 71$

Substitute the known values into the formula for the inverse.

$\frac{1}{- 71} [\begin{array}{c} - 4 & - 5 \\ - 7 & 9 \end{array}]$

Move the negative in front of the fraction.

$- \frac{1}{71} [\begin{array}{c} - 4 & - 5 \\ - 7 & 9 \end{array}]$

Multiply $- \frac{1}{71}$ by each element of the matrix.

$[\begin{array}{c} - \frac{1}{71} \cdot - 4 & - \frac{1}{71} \cdot - 5 \\ - \frac{1}{71} \cdot - 7 & - \frac{1}{71} \cdot 9 \end{array}]$

Simplify each element in the matrix.

Multiply $- \frac{1}{71} \cdot - 4$ .

Multiply $- 4$ by $- 1$ .

$[\begin{array}{c} 4 (\frac{1}{71}) & - \frac{1}{71} \cdot - 5 \\ - \frac{1}{71} \cdot - 7 & - \frac{1}{71} \cdot 9 \end{array}]$

Combine $4$ and $\frac{1}{71}$ .

$[\begin{array}{c} \frac{4}{71} & - \frac{1}{71} \cdot - 5 \\ - \frac{1}{71} \cdot - 7 & - \frac{1}{71} \cdot 9 \end{array}]$

Multiply $- \frac{1}{71} \cdot - 5$ .

Multiply $- 5$ by $- 1$ .

$[\begin{array}{c} \frac{4}{71} & 5 (\frac{1}{71}) \\ - \frac{1}{71} \cdot - 7 & - \frac{1}{71} \cdot 9 \end{array}]$

Combine $5$ and $\frac{1}{71}$ .

$[\begin{array}{c} \frac{4}{71} & \frac{5}{71} \\ - \frac{1}{71} \cdot - 7 & - \frac{1}{71} \cdot 9 \end{array}]$

Multiply $- \frac{1}{71} \cdot - 7$ .

Multiply $- 7$ by $- 1$ .

$[\begin{array}{c} \frac{4}{71} & \frac{5}{71} \\ 7 (\frac{1}{71}) & - \frac{1}{71} \cdot 9 \end{array}]$

Combine $7$ and $\frac{1}{71}$ .

$[\begin{array}{c} \frac{4}{71} & \frac{5}{71} \\ \frac{7}{71} & - \frac{1}{71} \cdot 9 \end{array}]$

Multiply $- \frac{1}{71} \cdot 9$ .

Multiply $9$ by $- 1$ .

$[\begin{array}{c} \frac{4}{71} & \frac{5}{71} \\ \frac{7}{71} & - 9 (\frac{1}{71}) \end{array}]$

Combine $- 9$ and $\frac{1}{71}$ .

$[\begin{array}{c} \frac{4}{71} & \frac{5}{71} \\ \frac{7}{71} & \frac{- 9}{71} \end{array}]$

Move the negative in front of the fraction.

$[\begin{array}{c} \frac{4}{71} & \frac{5}{71} \\ \frac{7}{71} & - \frac{9}{71} \end{array}]$

Step 2

Multiply both sides by the inverse of $[\begin{array}{c} 9 & 5 \\ 7 & - 4 \end{array}]$ .

$[\begin{array}{c} \frac{4}{71} & \frac{5}{71} \\ \frac{7}{71} & - \frac{9}{71} \end{array}] [\begin{array}{c} 9 & 5 \\ 7 & - 4 \end{array}] x = [\begin{array}{c} \frac{4}{71} & \frac{5}{71} \\ \frac{7}{71} & - \frac{9}{71} \end{array}] [\begin{array}{c} 2 \\ 1 \end{array}]$

Step 3

Simplify the equation.

Multiply $[\begin{array}{c} \frac{4}{71} & \frac{5}{71} \\ \frac{7}{71} & - \frac{9}{71} \end{array}] [\begin{array}{c} 9 & 5 \\ 7 & - 4 \end{array}]$ .

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

$[\begin{array}{c} \frac{4}{71} \cdot 9 + \frac{5}{71} \cdot 7 & \frac{4}{71} \cdot 5 + \frac{5}{71} \cdot - 4 \\ \frac{7}{71} \cdot 9 - \frac{9}{71} \cdot 7 & \frac{7}{71} \cdot 5 - \frac{9}{71} \cdot - 4 \end{array}] x = [\begin{array}{c} \frac{4}{71} & \frac{5}{71} \\ \frac{7}{71} & - \frac{9}{71} \end{array}] [\begin{array}{c} 2 \\ 1 \end{array}]$

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

$[\begin{array}{c} 1 & 0 \\ 0 & 1 \end{array}] x = [\begin{array}{c} \frac{4}{71} & \frac{5}{71} \\ \frac{7}{71} & - \frac{9}{71} \end{array}] [\begin{array}{c} 2 \\ 1 \end{array}]$

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

$x = [\begin{array}{c} \frac{4}{71} & \frac{5}{71} \\ \frac{7}{71} & - \frac{9}{71} \end{array}] [\begin{array}{c} 2 \\ 1 \end{array}]$

Multiply $[\begin{array}{c} \frac{4}{71} & \frac{5}{71} \\ \frac{7}{71} & - \frac{9}{71} \end{array}] [\begin{array}{c} 2 \\ 1 \end{array}]$ .

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

$x = [\begin{array}{c} \frac{4}{71} \cdot 2 + \frac{5}{71} \cdot 1 \\ \frac{7}{71} \cdot 2 - \frac{9}{71} \cdot 1 \end{array}]$

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

$x = [\begin{array}{c} \frac{13}{71} \\ \frac{5}{71} \end{array}]$

Solve the Matrix Equation [[9,5],[7,-4]]x=[[2],[1]]

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