Solve the Matrix Equation [[6,-5],[5,4]]X=[[3],[4]]

$[\begin{array}{c} 6 & - 5 \\ 5 & 4 \end{array}] X = [\begin{array}{c} 3 \\ 4 \end{array}]$

Step 1

Find the inverse of $[\begin{array}{c} 6 & - 5 \\ 5 & 4 \end{array}]$ .

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

$6 \cdot 4 - 5 \cdot - 5$

Simplify the determinant.

Simplify each term.

Multiply $6$ by $4$ .

$24 - 5 \cdot - 5$

Multiply $- 5$ by $- 5$ .

$24 + 25$

Add $24$ and $25$ .

$49$

Substitute the known values into the formula for the inverse.

$\frac{1}{49} [\begin{array}{c} 4 & 5 \\ - 5 & 6 \end{array}]$

Multiply $\frac{1}{49}$ by each element of the matrix.

$[\begin{array}{c} \frac{1}{49} \cdot 4 & \frac{1}{49} \cdot 5 \\ \frac{1}{49} \cdot - 5 & \frac{1}{49} \cdot 6 \end{array}]$

Simplify each element in the matrix.

Combine $\frac{1}{49}$ and $4$ .

$[\begin{array}{c} \frac{4}{49} & \frac{1}{49} \cdot 5 \\ \frac{1}{49} \cdot - 5 & \frac{1}{49} \cdot 6 \end{array}]$

Combine $\frac{1}{49}$ and $5$ .

$[\begin{array}{c} \frac{4}{49} & \frac{5}{49} \\ \frac{1}{49} \cdot - 5 & \frac{1}{49} \cdot 6 \end{array}]$

Combine $\frac{1}{49}$ and $- 5$ .

$[\begin{array}{c} \frac{4}{49} & \frac{5}{49} \\ \frac{- 5}{49} & \frac{1}{49} \cdot 6 \end{array}]$

Move the negative in front of the fraction.

$[\begin{array}{c} \frac{4}{49} & \frac{5}{49} \\ - \frac{5}{49} & \frac{1}{49} \cdot 6 \end{array}]$

Combine $\frac{1}{49}$ and $6$ .

$[\begin{array}{c} \frac{4}{49} & \frac{5}{49} \\ - \frac{5}{49} & \frac{6}{49} \end{array}]$

Step 2

Multiply both sides by the inverse of $[\begin{array}{c} 6 & - 5 \\ 5 & 4 \end{array}]$ .

$[\begin{array}{c} \frac{4}{49} & \frac{5}{49} \\ - \frac{5}{49} & \frac{6}{49} \end{array}] [\begin{array}{c} 6 & - 5 \\ 5 & 4 \end{array}] X = [\begin{array}{c} \frac{4}{49} & \frac{5}{49} \\ - \frac{5}{49} & \frac{6}{49} \end{array}] [\begin{array}{c} 3 \\ 4 \end{array}]$

Step 3

Simplify the equation.

Multiply $[\begin{array}{c} \frac{4}{49} & \frac{5}{49} \\ - \frac{5}{49} & \frac{6}{49} \end{array}] [\begin{array}{c} 6 & - 5 \\ 5 & 4 \end{array}]$ .

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

$[\begin{array}{c} \frac{4}{49} \cdot 6 + \frac{5}{49} \cdot 5 & \frac{4}{49} \cdot - 5 + \frac{5}{49} \cdot 4 \\ - \frac{5}{49} \cdot 6 + \frac{6}{49} \cdot 5 & - \frac{5}{49} \cdot - 5 + \frac{6}{49} \cdot 4 \end{array}] X = [\begin{array}{c} \frac{4}{49} & \frac{5}{49} \\ - \frac{5}{49} & \frac{6}{49} \end{array}] [\begin{array}{c} 3 \\ 4 \end{array}]$

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

$[\begin{array}{c} 1 & 0 \\ 0 & 1 \end{array}] X = [\begin{array}{c} \frac{4}{49} & \frac{5}{49} \\ - \frac{5}{49} & \frac{6}{49} \end{array}] [\begin{array}{c} 3 \\ 4 \end{array}]$

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

$X = [\begin{array}{c} \frac{4}{49} & \frac{5}{49} \\ - \frac{5}{49} & \frac{6}{49} \end{array}] [\begin{array}{c} 3 \\ 4 \end{array}]$

Multiply $[\begin{array}{c} \frac{4}{49} & \frac{5}{49} \\ - \frac{5}{49} & \frac{6}{49} \end{array}] [\begin{array}{c} 3 \\ 4 \end{array}]$ .

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

$X = [\begin{array}{c} \frac{4}{49} \cdot 3 + \frac{5}{49} \cdot 4 \\ - \frac{5}{49} \cdot 3 + \frac{6}{49} \cdot 4 \end{array}]$

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

$X = [\begin{array}{c} \frac{32}{49} \\ \frac{9}{49} \end{array}]$

Solve the Matrix Equation [[6,-5],[5,4]]X=[[3],[4]]

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