Solve the Matrix Equation 4x+[[6,-6],[-8,0],[11,7]]=[[11,0],[0,-2],[6,7]]

$4 x + [\begin{array}{c} 6 & - 6 \\ - 8 & 0 \\ 11 & 7 \end{array}] = [\begin{array}{c} 11 & 0 \\ 0 & - 2 \\ 6 & 7 \end{array}]$

Step 1

Subtract the matrix from both sides of the equation.

$4 X + [\begin{array}{c} 6 & - 6 \\ - 8 & 0 \\ 11 & 7 \end{array}] - [\begin{array}{c} 6 & - 6 \\ - 8 & 0 \\ 11 & 7 \end{array}] = [\begin{array}{c} 11 & 0 \\ 0 & - 2 \\ 6 & 7 \end{array}] - [\begin{array}{c} 6 & - 6 \\ - 8 & 0 \\ 11 & 7 \end{array}]$

Step 2

Simplify both sides of the equation.

Simplify the left side of the equation.

Subtract the corresponding elements.

$[\begin{array}{c} 6 - 6 & - 6 - (- 6) \\ - 8 - (- 8) & 0 - 0 \\ 11 - 11 & 7 - 7 \end{array}] + 4 X = [\begin{array}{c} 11 & 0 \\ 0 & - 2 \\ 6 & 7 \end{array}] - [\begin{array}{c} 6 & - 6 \\ - 8 & 0 \\ 11 & 7 \end{array}]$

Simplify each element of the matrix $[\begin{array}{c} 6 - 6 & - 6 - (- 6) \\ - 8 - (- 8) & 0 - 0 \\ 11 - 11 & 7 - 7 \end{array}]$ .

Simplify $6 - 6$ .

$[\begin{array}{c} 0 & - 6 - (- 6) \\ - 8 - (- 8) & 0 - 0 \\ 11 - 11 & 7 - 7 \end{array}] + 4 X = [\begin{array}{c} 11 & 0 \\ 0 & - 2 \\ 6 & 7 \end{array}] - [\begin{array}{c} 6 & - 6 \\ - 8 & 0 \\ 11 & 7 \end{array}]$

Simplify $- 6 - (- 6)$ .

$[\begin{array}{c} 0 & 0 \\ - 8 - (- 8) & 0 - 0 \\ 11 - 11 & 7 - 7 \end{array}] + 4 X = [\begin{array}{c} 11 & 0 \\ 0 & - 2 \\ 6 & 7 \end{array}] - [\begin{array}{c} 6 & - 6 \\ - 8 & 0 \\ 11 & 7 \end{array}]$

Simplify $- 8 - (- 8)$ .

$[\begin{array}{c} 0 & 0 \\ 0 & 0 - 0 \\ 11 - 11 & 7 - 7 \end{array}] + 4 X = [\begin{array}{c} 11 & 0 \\ 0 & - 2 \\ 6 & 7 \end{array}] - [\begin{array}{c} 6 & - 6 \\ - 8 & 0 \\ 11 & 7 \end{array}]$

Simplify $0 - 0$ .

$[\begin{array}{c} 0 & 0 \\ 0 & 0 \\ 11 - 11 & 7 - 7 \end{array}] + 4 X = [\begin{array}{c} 11 & 0 \\ 0 & - 2 \\ 6 & 7 \end{array}] - [\begin{array}{c} 6 & - 6 \\ - 8 & 0 \\ 11 & 7 \end{array}]$

Simplify $11 - 11$ .

$[\begin{array}{c} 0 & 0 \\ 0 & 0 \\ 0 & 7 - 7 \end{array}] + 4 X = [\begin{array}{c} 11 & 0 \\ 0 & - 2 \\ 6 & 7 \end{array}] - [\begin{array}{c} 6 & - 6 \\ - 8 & 0 \\ 11 & 7 \end{array}]$

Simplify $7 - 7$ .

$[\begin{array}{c} 0 & 0 \\ 0 & 0 \\ 0 & 0 \end{array}] + 4 X = [\begin{array}{c} 11 & 0 \\ 0 & - 2 \\ 6 & 7 \end{array}] - [\begin{array}{c} 6 & - 6 \\ - 8 & 0 \\ 11 & 7 \end{array}]$

Simplify the right side of the equation.

Subtract the corresponding elements.

$[\begin{array}{c} 0 & 0 \\ 0 & 0 \\ 0 & 0 \end{array}] + 4 X = [\begin{array}{c} 11 - 6 & 0 - (- 6) \\ 0 - (- 8) & - 2 - 0 \\ 6 - 11 & 7 - 7 \end{array}]$

Simplify each element of the matrix $[\begin{array}{c} 11 - 6 & 0 - (- 6) \\ 0 - (- 8) & - 2 - 0 \\ 6 - 11 & 7 - 7 \end{array}]$ .

Simplify $11 - 6$ .

$[\begin{array}{c} 0 & 0 \\ 0 & 0 \\ 0 & 0 \end{array}] + 4 X = [\begin{array}{c} 5 & 0 - (- 6) \\ 0 - (- 8) & - 2 - 0 \\ 6 - 11 & 7 - 7 \end{array}]$

Simplify $0 - (- 6)$ .

$[\begin{array}{c} 0 & 0 \\ 0 & 0 \\ 0 & 0 \end{array}] + 4 X = [\begin{array}{c} 5 & 6 \\ 0 - (- 8) & - 2 - 0 \\ 6 - 11 & 7 - 7 \end{array}]$

Simplify $0 - (- 8)$ .

$[\begin{array}{c} 0 & 0 \\ 0 & 0 \\ 0 & 0 \end{array}] + 4 X = [\begin{array}{c} 5 & 6 \\ 8 & - 2 - 0 \\ 6 - 11 & 7 - 7 \end{array}]$

Simplify $- 2 - 0$ .

$[\begin{array}{c} 0 & 0 \\ 0 & 0 \\ 0 & 0 \end{array}] + 4 X = [\begin{array}{c} 5 & 6 \\ 8 & - 2 \\ 6 - 11 & 7 - 7 \end{array}]$

Simplify $6 - 11$ .

$[\begin{array}{c} 0 & 0 \\ 0 & 0 \\ 0 & 0 \end{array}] + 4 X = [\begin{array}{c} 5 & 6 \\ 8 & - 2 \\ - 5 & 7 - 7 \end{array}]$

Simplify $7 - 7$ .

$[\begin{array}{c} 0 & 0 \\ 0 & 0 \\ 0 & 0 \end{array}] + 4 X = [\begin{array}{c} 5 & 6 \\ 8 & - 2 \\ - 5 & 0 \end{array}]$

Adding the null matrix to any matrix is a matrix itself.

$4 X = [\begin{array}{c} 5 & 6 \\ 8 & - 2 \\ - 5 & 0 \end{array}]$

Step 3

Multiply both sides of the equation by $\frac{1}{4}$ .

$\frac{1}{4} \cdot (4 X) = \frac{1}{4} \cdot [\begin{array}{c} 5 & 6 \\ 8 & - 2 \\ - 5 & 0 \end{array}]$

Step 4

Simplify both sides of the equation.

Cancel the common factor of $4$ .

Factor $4$ out of $4 X$ .

$\frac{1}{4} \cdot (4 (X)) = \frac{1}{4} \cdot [\begin{array}{c} 5 & 6 \\ 8 & - 2 \\ - 5 & 0 \end{array}]$

Cancel the common factor.

$\frac{1}{4} \cdot (4 X) = \frac{1}{4} \cdot [\begin{array}{c} 5 & 6 \\ 8 & - 2 \\ - 5 & 0 \end{array}]$

Rewrite the expression.

$X = \frac{1}{4} \cdot [\begin{array}{c} 5 & 6 \\ 8 & - 2 \\ - 5 & 0 \end{array}]$

Simplify the right side of the equation.

Multiply $\frac{1}{4}$ by each element of the matrix.

$X = [\begin{array}{c} \frac{1}{4} \cdot 5 & \frac{1}{4} \cdot 6 \\ \frac{1}{4} \cdot 8 & \frac{1}{4} \cdot - 2 \\ \frac{1}{4} \cdot - 5 & \frac{1}{4} \cdot 0 \end{array}]$

Simplify each element in the matrix.

Rearrange $\frac{1}{4} \cdot 5$ .

$X = [\begin{array}{c} \frac{5}{4} & \frac{1}{4} \cdot 6 \\ \frac{1}{4} \cdot 8 & \frac{1}{4} \cdot - 2 \\ \frac{1}{4} \cdot - 5 & \frac{1}{4} \cdot 0 \end{array}]$

Rearrange $\frac{1}{4} \cdot 6$ .

$X = [\begin{array}{c} \frac{5}{4} & \frac{3}{2} \\ \frac{1}{4} \cdot 8 & \frac{1}{4} \cdot - 2 \\ \frac{1}{4} \cdot - 5 & \frac{1}{4} \cdot 0 \end{array}]$

Rearrange $\frac{1}{4} \cdot 8$ .

$X = [\begin{array}{c} \frac{5}{4} & \frac{3}{2} \\ 2 & \frac{1}{4} \cdot - 2 \\ \frac{1}{4} \cdot - 5 & \frac{1}{4} \cdot 0 \end{array}]$

Rearrange $\frac{1}{4} \cdot - 2$ .

$X = [\begin{array}{c} \frac{5}{4} & \frac{3}{2} \\ 2 & - \frac{1}{2} \\ \frac{1}{4} \cdot - 5 & \frac{1}{4} \cdot 0 \end{array}]$

Rearrange $\frac{1}{4} \cdot - 5$ .

$X = [\begin{array}{c} \frac{5}{4} & \frac{3}{2} \\ 2 & - \frac{1}{2} \\ - \frac{5}{4} & \frac{1}{4} \cdot 0 \end{array}]$

Rearrange $\frac{1}{4} \cdot 0$ .

$X = [\begin{array}{c} \frac{5}{4} & \frac{3}{2} \\ 2 & - \frac{1}{2} \\ - \frac{5}{4} & 0 \end{array}]$

Solve the Matrix Equation 4x+[[6,-6],[-8,0],[11,7]]=[[11,0],[0,-2],[6,7]]

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