Solve the Matrix Equation -5X+[[5,5],[2,-1],[-8,16]]=[[10,-25],[17,-31],[-23,11]]

$- 5 X + [\begin{array}{c} 5 & 5 \\ 2 & - 1 \\ - 8 & 16 \end{array}] = [\begin{array}{c} 10 & - 25 \\ 17 & - 31 \\ - 23 & 11 \end{array}]$

Subtract the matrix from both sides of the equation.

$- 5 X + [\begin{array}{c} 5 & 5 \\ 2 & - 1 \\ - 8 & 16 \end{array}] - [\begin{array}{c} 5 & 5 \\ 2 & - 1 \\ - 8 & 16 \end{array}] = [\begin{array}{c} 10 & - 25 \\ 17 & - 31 \\ - 23 & 11 \end{array}] - [\begin{array}{c} 5 & 5 \\ 2 & - 1 \\ - 8 & 16 \end{array}]$

Simplify both sides of the equation.

Simplify the left side of the equation.

Subtract the corresponding elements.

$[\begin{array}{c} 5 - 5 & 5 - 5 \\ 2 - 2 & - 1 - (- 1) \\ - 8 - (- 8) & 16 - 16 \end{array}] - 5 X = [\begin{array}{c} 10 & - 25 \\ 17 & - 31 \\ - 23 & 11 \end{array}] - [\begin{array}{c} 5 & 5 \\ 2 & - 1 \\ - 8 & 16 \end{array}]$

Simplify each element of the matrix $[\begin{array}{c} 5 - 5 & 5 - 5 \\ 2 - 2 & - 1 - (- 1) \\ - 8 - (- 8) & 16 - 16 \end{array}]$ .

Simplify $5 - 5$ .

$[\begin{array}{c} 0 & 5 - 5 \\ 2 - 2 & - 1 - (- 1) \\ - 8 - (- 8) & 16 - 16 \end{array}] - 5 X = [\begin{array}{c} 10 & - 25 \\ 17 & - 31 \\ - 23 & 11 \end{array}] - [\begin{array}{c} 5 & 5 \\ 2 & - 1 \\ - 8 & 16 \end{array}]$

Simplify $5 - 5$ .

$[\begin{array}{c} 0 & 0 \\ 2 - 2 & - 1 - (- 1) \\ - 8 - (- 8) & 16 - 16 \end{array}] - 5 X = [\begin{array}{c} 10 & - 25 \\ 17 & - 31 \\ - 23 & 11 \end{array}] - [\begin{array}{c} 5 & 5 \\ 2 & - 1 \\ - 8 & 16 \end{array}]$

Simplify $2 - 2$ .

$[\begin{array}{c} 0 & 0 \\ 0 & - 1 - (- 1) \\ - 8 - (- 8) & 16 - 16 \end{array}] - 5 X = [\begin{array}{c} 10 & - 25 \\ 17 & - 31 \\ - 23 & 11 \end{array}] - [\begin{array}{c} 5 & 5 \\ 2 & - 1 \\ - 8 & 16 \end{array}]$

Simplify $- 1 - (- 1)$ .

$[\begin{array}{c} 0 & 0 \\ 0 & 0 \\ - 8 - (- 8) & 16 - 16 \end{array}] - 5 X = [\begin{array}{c} 10 & - 25 \\ 17 & - 31 \\ - 23 & 11 \end{array}] - [\begin{array}{c} 5 & 5 \\ 2 & - 1 \\ - 8 & 16 \end{array}]$

Simplify $- 8 - (- 8)$ .

$[\begin{array}{c} 0 & 0 \\ 0 & 0 \\ 0 & 16 - 16 \end{array}] - 5 X = [\begin{array}{c} 10 & - 25 \\ 17 & - 31 \\ - 23 & 11 \end{array}] - [\begin{array}{c} 5 & 5 \\ 2 & - 1 \\ - 8 & 16 \end{array}]$

Simplify $16 - 16$ .

$[\begin{array}{c} 0 & 0 \\ 0 & 0 \\ 0 & 0 \end{array}] - 5 X = [\begin{array}{c} 10 & - 25 \\ 17 & - 31 \\ - 23 & 11 \end{array}] - [\begin{array}{c} 5 & 5 \\ 2 & - 1 \\ - 8 & 16 \end{array}]$

Simplify the right side of the equation.

Subtract the corresponding elements.

$[\begin{array}{c} 0 & 0 \\ 0 & 0 \\ 0 & 0 \end{array}] - 5 X = [\begin{array}{c} 10 - 5 & - 25 - 5 \\ 17 - 2 & - 31 - (- 1) \\ - 23 - (- 8) & 11 - 16 \end{array}]$

Simplify each element of the matrix $[\begin{array}{c} 10 - 5 & - 25 - 5 \\ 17 - 2 & - 31 - (- 1) \\ - 23 - (- 8) & 11 - 16 \end{array}]$ .

Simplify $10 - 5$ .

$[\begin{array}{c} 0 & 0 \\ 0 & 0 \\ 0 & 0 \end{array}] - 5 X = [\begin{array}{c} 5 & - 25 - 5 \\ 17 - 2 & - 31 - (- 1) \\ - 23 - (- 8) & 11 - 16 \end{array}]$

Simplify $- 25 - 5$ .

$[\begin{array}{c} 0 & 0 \\ 0 & 0 \\ 0 & 0 \end{array}] - 5 X = [\begin{array}{c} 5 & - 30 \\ 17 - 2 & - 31 - (- 1) \\ - 23 - (- 8) & 11 - 16 \end{array}]$

Simplify $17 - 2$ .

$[\begin{array}{c} 0 & 0 \\ 0 & 0 \\ 0 & 0 \end{array}] - 5 X = [\begin{array}{c} 5 & - 30 \\ 15 & - 31 - (- 1) \\ - 23 - (- 8) & 11 - 16 \end{array}]$

Simplify $- 31 - (- 1)$ .

$[\begin{array}{c} 0 & 0 \\ 0 & 0 \\ 0 & 0 \end{array}] - 5 X = [\begin{array}{c} 5 & - 30 \\ 15 & - 30 \\ - 23 - (- 8) & 11 - 16 \end{array}]$

Simplify $- 23 - (- 8)$ .

$[\begin{array}{c} 0 & 0 \\ 0 & 0 \\ 0 & 0 \end{array}] - 5 X = [\begin{array}{c} 5 & - 30 \\ 15 & - 30 \\ - 15 & 11 - 16 \end{array}]$

Simplify $11 - 16$ .

$[\begin{array}{c} 0 & 0 \\ 0 & 0 \\ 0 & 0 \end{array}] - 5 X = [\begin{array}{c} 5 & - 30 \\ 15 & - 30 \\ - 15 & - 5 \end{array}]$

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

$- 5 X = [\begin{array}{c} 5 & - 30 \\ 15 & - 30 \\ - 15 & - 5 \end{array}]$

Multiply both sides of the equation by $- \frac{1}{5}$ .

$- \frac{1}{5} \cdot (- 5 X) = - \frac{1}{5} \cdot [\begin{array}{c} 5 & - 30 \\ 15 & - 30 \\ - 15 & - 5 \end{array}]$

Simplify both sides of the equation.

Simplify the left side of the equation.

Cancel the common factor of $5$ .

Move the leading negative in $- \frac{1}{5}$ into the numerator.

$\frac{- 1}{5} \cdot (- 5 X) = - \frac{1}{5} \cdot [\begin{array}{c} 5 & - 30 \\ 15 & - 30 \\ - 15 & - 5 \end{array}]$

Factor $5$ out of $- 5 X$ .

$\frac{- 1}{5} \cdot (5 (- X)) = - \frac{1}{5} \cdot [\begin{array}{c} 5 & - 30 \\ 15 & - 30 \\ - 15 & - 5 \end{array}]$

Cancel the common factor.

$\frac{- 1}{5} \cdot (5 (- X)) = - \frac{1}{5} \cdot [\begin{array}{c} 5 & - 30 \\ 15 & - 30 \\ - 15 & - 5 \end{array}]$

Rewrite the expression.

$- 1 \cdot (- X) = - \frac{1}{5} \cdot [\begin{array}{c} 5 & - 30 \\ 15 & - 30 \\ - 15 & - 5 \end{array}]$

Multiply.

Multiply $- 1$ by $- 1$ .

$1 \cdot X = - \frac{1}{5} \cdot [\begin{array}{c} 5 & - 30 \\ 15 & - 30 \\ - 15 & - 5 \end{array}]$

Multiply $X$ by $1$ .

$X = - \frac{1}{5} \cdot [\begin{array}{c} 5 & - 30 \\ 15 & - 30 \\ - 15 & - 5 \end{array}]$

Simplify the right side of the equation.

Multiply $- \frac{1}{5}$ by each element of the matrix.

$X = [\begin{array}{c} - \frac{1}{5} \cdot 5 & - \frac{1}{5} \cdot - 30 \\ - \frac{1}{5} \cdot 15 & - \frac{1}{5} \cdot - 30 \\ - \frac{1}{5} \cdot - 15 & - \frac{1}{5} \cdot - 5 \end{array}]$

Simplify each element in the matrix.

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

$X = [\begin{array}{c} - 1 & - \frac{1}{5} \cdot - 30 \\ - \frac{1}{5} \cdot 15 & - \frac{1}{5} \cdot - 30 \\ - \frac{1}{5} \cdot - 15 & - \frac{1}{5} \cdot - 5 \end{array}]$

Rearrange $- \frac{1}{5} \cdot - 30$ .

$X = [\begin{array}{c} - 1 & 6 \\ - \frac{1}{5} \cdot 15 & - \frac{1}{5} \cdot - 30 \\ - \frac{1}{5} \cdot - 15 & - \frac{1}{5} \cdot - 5 \end{array}]$

Rearrange $- \frac{1}{5} \cdot 15$ .

$X = [\begin{array}{c} - 1 & 6 \\ - 3 & - \frac{1}{5} \cdot - 30 \\ - \frac{1}{5} \cdot - 15 & - \frac{1}{5} \cdot - 5 \end{array}]$

Rearrange $- \frac{1}{5} \cdot - 30$ .

$X = [\begin{array}{c} - 1 & 6 \\ - 3 & 6 \\ - \frac{1}{5} \cdot - 15 & - \frac{1}{5} \cdot - 5 \end{array}]$

Rearrange $- \frac{1}{5} \cdot - 15$ .

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

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

$X = [\begin{array}{c} - 1 & 6 \\ - 3 & 6 \\ 3 & 1 \end{array}]$

Solve the Matrix Equation -5X+[[5,5],[2,-1],[-8,16]]=[[10,-25],[17,-31],[-23,11]]

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