Solve the Matrix Equation 2X+2[[2,-8],[-4,2]]=[[4,-6],[2,-8]]

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

Multiply $2$ by each element of the matrix.

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

Simplify each element in the matrix.

Rearrange $2 \cdot 2$ .

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

Rearrange $2 \cdot - 8$ .

$2 X + [\begin{array}{c} 4 & - 16 \\ 2 \cdot - 4 & 2 \cdot 2 \end{array}] = [\begin{array}{c} 4 & - 6 \\ 2 & - 8 \end{array}]$

Rearrange $2 \cdot - 4$ .

$2 X + [\begin{array}{c} 4 & - 16 \\ - 8 & 2 \cdot 2 \end{array}] = [\begin{array}{c} 4 & - 6 \\ 2 & - 8 \end{array}]$

Rearrange $2 \cdot 2$ .

$2 X + [\begin{array}{c} 4 & - 16 \\ - 8 & 4 \end{array}] = [\begin{array}{c} 4 & - 6 \\ 2 & - 8 \end{array}]$

Subtract the matrix from both sides of the equation.

$2 X + [\begin{array}{c} 4 & - 16 \\ - 8 & 4 \end{array}] - [\begin{array}{c} 4 & - 16 \\ - 8 & 4 \end{array}] = [\begin{array}{c} 4 & - 6 \\ 2 & - 8 \end{array}] - [\begin{array}{c} 4 & - 16 \\ - 8 & 4 \end{array}]$

Simplify both sides of the equation.

Simplify the left side of the equation.

Subtract the corresponding elements.

$[\begin{array}{c} 4 - 4 & - 16 - (- 16) \\ - 8 - (- 8) & 4 - 4 \end{array}] + 2 X = [\begin{array}{c} 4 & - 6 \\ 2 & - 8 \end{array}] - [\begin{array}{c} 4 & - 16 \\ - 8 & 4 \end{array}]$

Simplify each element of the matrix $[\begin{array}{c} 4 - 4 & - 16 - (- 16) \\ - 8 - (- 8) & 4 - 4 \end{array}]$ .

Simplify $4 - 4$ .

$[\begin{array}{c} 0 & - 16 - (- 16) \\ - 8 - (- 8) & 4 - 4 \end{array}] + 2 X = [\begin{array}{c} 4 & - 6 \\ 2 & - 8 \end{array}] - [\begin{array}{c} 4 & - 16 \\ - 8 & 4 \end{array}]$

Simplify $- 16 - (- 16)$ .

$[\begin{array}{c} 0 & 0 \\ - 8 - (- 8) & 4 - 4 \end{array}] + 2 X = [\begin{array}{c} 4 & - 6 \\ 2 & - 8 \end{array}] - [\begin{array}{c} 4 & - 16 \\ - 8 & 4 \end{array}]$

Simplify $- 8 - (- 8)$ .

$[\begin{array}{c} 0 & 0 \\ 0 & 4 - 4 \end{array}] + 2 X = [\begin{array}{c} 4 & - 6 \\ 2 & - 8 \end{array}] - [\begin{array}{c} 4 & - 16 \\ - 8 & 4 \end{array}]$

Simplify $4 - 4$ .

$[\begin{array}{c} 0 & 0 \\ 0 & 0 \end{array}] + 2 X = [\begin{array}{c} 4 & - 6 \\ 2 & - 8 \end{array}] - [\begin{array}{c} 4 & - 16 \\ - 8 & 4 \end{array}]$

Simplify the right side of the equation.

Subtract the corresponding elements.

$[\begin{array}{c} 0 & 0 \\ 0 & 0 \end{array}] + 2 X = [\begin{array}{c} 4 - 4 & - 6 - (- 16) \\ 2 - (- 8) & - 8 - 4 \end{array}]$

Simplify each element of the matrix $[\begin{array}{c} 4 - 4 & - 6 - (- 16) \\ 2 - (- 8) & - 8 - 4 \end{array}]$ .

Simplify $4 - 4$ .

$[\begin{array}{c} 0 & 0 \\ 0 & 0 \end{array}] + 2 X = [\begin{array}{c} 0 & - 6 - (- 16) \\ 2 - (- 8) & - 8 - 4 \end{array}]$

Simplify $- 6 - (- 16)$ .

$[\begin{array}{c} 0 & 0 \\ 0 & 0 \end{array}] + 2 X = [\begin{array}{c} 0 & 10 \\ 2 - (- 8) & - 8 - 4 \end{array}]$

Simplify $2 - (- 8)$ .

$[\begin{array}{c} 0 & 0 \\ 0 & 0 \end{array}] + 2 X = [\begin{array}{c} 0 & 10 \\ 10 & - 8 - 4 \end{array}]$

Simplify $- 8 - 4$ .

$[\begin{array}{c} 0 & 0 \\ 0 & 0 \end{array}] + 2 X = [\begin{array}{c} 0 & 10 \\ 10 & - 12 \end{array}]$

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

$2 X = [\begin{array}{c} 0 & 10 \\ 10 & - 12 \end{array}]$

Multiply both sides of the equation by $\frac{1}{2}$ .

$\frac{1}{2} \cdot (2 X) = \frac{1}{2} \cdot [\begin{array}{c} 0 & 10 \\ 10 & - 12 \end{array}]$

Simplify both sides of the equation.

Cancel the common factor of $2$ .

Factor $2$ out of $2 X$ .

$\frac{1}{2} \cdot (2 (X)) = \frac{1}{2} \cdot [\begin{array}{c} 0 & 10 \\ 10 & - 12 \end{array}]$

Cancel the common factor.

$\frac{1}{2} \cdot (2 X) = \frac{1}{2} \cdot [\begin{array}{c} 0 & 10 \\ 10 & - 12 \end{array}]$

Rewrite the expression.

$X = \frac{1}{2} \cdot [\begin{array}{c} 0 & 10 \\ 10 & - 12 \end{array}]$

Simplify the right side of the equation.

Multiply $\frac{1}{2}$ by each element of the matrix.

$X = [\begin{array}{c} \frac{1}{2} \cdot 0 & \frac{1}{2} \cdot 10 \\ \frac{1}{2} \cdot 10 & \frac{1}{2} \cdot - 12 \end{array}]$

Simplify each element in the matrix.

Rearrange $\frac{1}{2} \cdot 0$ .

$X = [\begin{array}{c} 0 & \frac{1}{2} \cdot 10 \\ \frac{1}{2} \cdot 10 & \frac{1}{2} \cdot - 12 \end{array}]$

Rearrange $\frac{1}{2} \cdot 10$ .

$X = [\begin{array}{c} 0 & 5 \\ \frac{1}{2} \cdot 10 & \frac{1}{2} \cdot - 12 \end{array}]$

Rearrange $\frac{1}{2} \cdot 10$ .

$X = [\begin{array}{c} 0 & 5 \\ 5 & \frac{1}{2} \cdot - 12 \end{array}]$

Rearrange $\frac{1}{2} \cdot - 12$ .

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

Solve the Matrix Equation 2X+2[[2,-8],[-4,2]]=[[4,-6],[2,-8]]

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