Solve the Matrix Equation -2x+[[14,12,9],[7,8,2]]=[[-8,32,-13],[19,-14,-6]]

$- 2 x + [\begin{array}{c} 14 & 12 & 9 \\ 7 & 8 & 2 \end{array}] = [\begin{array}{c} - 8 & 32 & - 13 \\ 19 & - 14 & - 6 \end{array}]$

Subtract the matrix from both sides of the equation.

$- 2 X + [\begin{array}{c} 14 & 12 & 9 \\ 7 & 8 & 2 \end{array}] - [\begin{array}{c} 14 & 12 & 9 \\ 7 & 8 & 2 \end{array}] = [\begin{array}{c} - 8 & 32 & - 13 \\ 19 & - 14 & - 6 \end{array}] - [\begin{array}{c} 14 & 12 & 9 \\ 7 & 8 & 2 \end{array}]$

Simplify both sides of the equation.

Simplify the left side of the equation.

Subtract the corresponding elements.

$[\begin{array}{c} 14 - 14 & 12 - 12 & 9 - 9 \\ 7 - 7 & 8 - 8 & 2 - 2 \end{array}] - 2 X = [\begin{array}{c} - 8 & 32 & - 13 \\ 19 & - 14 & - 6 \end{array}] - [\begin{array}{c} 14 & 12 & 9 \\ 7 & 8 & 2 \end{array}]$

Simplify each element of the matrix $[\begin{array}{c} 14 - 14 & 12 - 12 & 9 - 9 \\ 7 - 7 & 8 - 8 & 2 - 2 \end{array}]$ .

Simplify $14 - 14$ .

$[\begin{array}{c} 0 & 12 - 12 & 9 - 9 \\ 7 - 7 & 8 - 8 & 2 - 2 \end{array}] - 2 X = [\begin{array}{c} - 8 & 32 & - 13 \\ 19 & - 14 & - 6 \end{array}] - [\begin{array}{c} 14 & 12 & 9 \\ 7 & 8 & 2 \end{array}]$

Simplify $12 - 12$ .

$[\begin{array}{c} 0 & 0 & 9 - 9 \\ 7 - 7 & 8 - 8 & 2 - 2 \end{array}] - 2 X = [\begin{array}{c} - 8 & 32 & - 13 \\ 19 & - 14 & - 6 \end{array}] - [\begin{array}{c} 14 & 12 & 9 \\ 7 & 8 & 2 \end{array}]$

Simplify $9 - 9$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 7 - 7 & 8 - 8 & 2 - 2 \end{array}] - 2 X = [\begin{array}{c} - 8 & 32 & - 13 \\ 19 & - 14 & - 6 \end{array}] - [\begin{array}{c} 14 & 12 & 9 \\ 7 & 8 & 2 \end{array}]$

Simplify $7 - 7$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 8 - 8 & 2 - 2 \end{array}] - 2 X = [\begin{array}{c} - 8 & 32 & - 13 \\ 19 & - 14 & - 6 \end{array}] - [\begin{array}{c} 14 & 12 & 9 \\ 7 & 8 & 2 \end{array}]$

Simplify $8 - 8$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 0 & 2 - 2 \end{array}] - 2 X = [\begin{array}{c} - 8 & 32 & - 13 \\ 19 & - 14 & - 6 \end{array}] - [\begin{array}{c} 14 & 12 & 9 \\ 7 & 8 & 2 \end{array}]$

Simplify $2 - 2$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}] - 2 X = [\begin{array}{c} - 8 & 32 & - 13 \\ 19 & - 14 & - 6 \end{array}] - [\begin{array}{c} 14 & 12 & 9 \\ 7 & 8 & 2 \end{array}]$

Simplify the right side of the equation.

Subtract the corresponding elements.

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}] - 2 X = [\begin{array}{c} - 8 - 14 & 32 - 12 & - 13 - 9 \\ 19 - 7 & - 14 - 8 & - 6 - 2 \end{array}]$

Simplify each element of the matrix $[\begin{array}{c} - 8 - 14 & 32 - 12 & - 13 - 9 \\ 19 - 7 & - 14 - 8 & - 6 - 2 \end{array}]$ .

Simplify $- 8 - 14$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}] - 2 X = [\begin{array}{c} - 22 & 32 - 12 & - 13 - 9 \\ 19 - 7 & - 14 - 8 & - 6 - 2 \end{array}]$

Simplify $32 - 12$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}] - 2 X = [\begin{array}{c} - 22 & 20 & - 13 - 9 \\ 19 - 7 & - 14 - 8 & - 6 - 2 \end{array}]$

Simplify $- 13 - 9$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}] - 2 X = [\begin{array}{c} - 22 & 20 & - 22 \\ 19 - 7 & - 14 - 8 & - 6 - 2 \end{array}]$

Simplify $19 - 7$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}] - 2 X = [\begin{array}{c} - 22 & 20 & - 22 \\ 12 & - 14 - 8 & - 6 - 2 \end{array}]$

Simplify $- 14 - 8$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}] - 2 X = [\begin{array}{c} - 22 & 20 & - 22 \\ 12 & - 22 & - 6 - 2 \end{array}]$

Simplify $- 6 - 2$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}] - 2 X = [\begin{array}{c} - 22 & 20 & - 22 \\ 12 & - 22 & - 8 \end{array}]$

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

$- 2 X = [\begin{array}{c} - 22 & 20 & - 22 \\ 12 & - 22 & - 8 \end{array}]$

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

$- \frac{1}{2} \cdot (- 2 X) = - \frac{1}{2} \cdot [\begin{array}{c} - 22 & 20 & - 22 \\ 12 & - 22 & - 8 \end{array}]$

Simplify both sides of the equation.

Simplify the left side of the equation.

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{1}{2}$ into the numerator.

$\frac{- 1}{2} \cdot (- 2 X) = - \frac{1}{2} \cdot [\begin{array}{c} - 22 & 20 & - 22 \\ 12 & - 22 & - 8 \end{array}]$

Factor $2$ out of $- 2 X$ .

$\frac{- 1}{2} \cdot (2 (- X)) = - \frac{1}{2} \cdot [\begin{array}{c} - 22 & 20 & - 22 \\ 12 & - 22 & - 8 \end{array}]$

Cancel the common factor.

$\frac{- 1}{2} \cdot (2 (- X)) = - \frac{1}{2} \cdot [\begin{array}{c} - 22 & 20 & - 22 \\ 12 & - 22 & - 8 \end{array}]$

Rewrite the expression.

$- 1 \cdot (- X) = - \frac{1}{2} \cdot [\begin{array}{c} - 22 & 20 & - 22 \\ 12 & - 22 & - 8 \end{array}]$

Multiply.

Multiply $- 1$ by $- 1$ .

$1 \cdot X = - \frac{1}{2} \cdot [\begin{array}{c} - 22 & 20 & - 22 \\ 12 & - 22 & - 8 \end{array}]$

Multiply $X$ by $1$ .

$X = - \frac{1}{2} \cdot [\begin{array}{c} - 22 & 20 & - 22 \\ 12 & - 22 & - 8 \end{array}]$

Simplify the right side of the equation.

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

$X = [\begin{array}{c} - \frac{1}{2} \cdot - 22 & - \frac{1}{2} \cdot 20 & - \frac{1}{2} \cdot - 22 \\ - \frac{1}{2} \cdot 12 & - \frac{1}{2} \cdot - 22 & - \frac{1}{2} \cdot - 8 \end{array}]$

Simplify each element in the matrix.

Rearrange $- \frac{1}{2} \cdot - 22$ .

$X = [\begin{array}{c} 11 & - \frac{1}{2} \cdot 20 & - \frac{1}{2} \cdot - 22 \\ - \frac{1}{2} \cdot 12 & - \frac{1}{2} \cdot - 22 & - \frac{1}{2} \cdot - 8 \end{array}]$

Rearrange $- \frac{1}{2} \cdot 20$ .

$X = [\begin{array}{c} 11 & - 10 & - \frac{1}{2} \cdot - 22 \\ - \frac{1}{2} \cdot 12 & - \frac{1}{2} \cdot - 22 & - \frac{1}{2} \cdot - 8 \end{array}]$

Rearrange $- \frac{1}{2} \cdot - 22$ .

$X = [\begin{array}{c} 11 & - 10 & 11 \\ - \frac{1}{2} \cdot 12 & - \frac{1}{2} \cdot - 22 & - \frac{1}{2} \cdot - 8 \end{array}]$

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

$X = [\begin{array}{c} 11 & - 10 & 11 \\ - 6 & - \frac{1}{2} \cdot - 22 & - \frac{1}{2} \cdot - 8 \end{array}]$

Rearrange $- \frac{1}{2} \cdot - 22$ .

$X = [\begin{array}{c} 11 & - 10 & 11 \\ - 6 & 11 & - \frac{1}{2} \cdot - 8 \end{array}]$

Rearrange $- \frac{1}{2} \cdot - 8$ .

$X = [\begin{array}{c} 11 & - 10 & 11 \\ - 6 & 11 & 4 \end{array}]$

Solve the Matrix Equation -2x+[[14,12,9],[7,8,2]]=[[-8,32,-13],[19,-14,-6]]

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