Solve the Matrix Equation -2x-[[8,4,4],[1,-5,-9],[3,2,0]]=[[-18,2,22],[9,-25,-17],[15,12,10]]

$- 2 x - [\begin{array}{c} 8 & 4 & 4 \\ 1 & - 5 & - 9 \\ 3 & 2 & 0 \end{array}] = [\begin{array}{c} - 18 & 2 & 22 \\ 9 & - 25 & - 17 \\ 15 & 12 & 10 \end{array}]$

Step 1

Add the matrix to both sides of the equation.

$- 2 X - [\begin{array}{c} 8 & 4 & 4 \\ 1 & - 5 & - 9 \\ 3 & 2 & 0 \end{array}] + [\begin{array}{c} 8 & 4 & 4 \\ 1 & - 5 & - 9 \\ 3 & 2 & 0 \end{array}] = [\begin{array}{c} - 18 & 2 & 22 \\ 9 & - 25 & - 17 \\ 15 & 12 & 10 \end{array}] + [\begin{array}{c} 8 & 4 & 4 \\ 1 & - 5 & - 9 \\ 3 & 2 & 0 \end{array}]$

Step 2

Simplify both sides of the equation.

Simplify the left side of the equation.

Add the corresponding elements.

$[\begin{array}{c} - 8 + 8 & - 4 + 4 & - 4 + 4 \\ - 1 + 1 & 5 - 5 & 9 - 9 \\ - 3 + 3 & - 2 + 2 & 0 + 0 \end{array}] - 2 X = [\begin{array}{c} - 18 & 2 & 22 \\ 9 & - 25 & - 17 \\ 15 & 12 & 10 \end{array}] + [\begin{array}{c} 8 & 4 & 4 \\ 1 & - 5 & - 9 \\ 3 & 2 & 0 \end{array}]$

Simplify each element of the matrix $[\begin{array}{c} - 8 + 8 & - 4 + 4 & - 4 + 4 \\ - 1 + 1 & 5 - 5 & 9 - 9 \\ - 3 + 3 & - 2 + 2 & 0 + 0 \end{array}]$ .

Simplify $- 8 + 8$ .

$[\begin{array}{c} 0 & - 4 + 4 & - 4 + 4 \\ - 1 + 1 & 5 - 5 & 9 - 9 \\ - 3 + 3 & - 2 + 2 & 0 + 0 \end{array}] - 2 X = [\begin{array}{c} - 18 & 2 & 22 \\ 9 & - 25 & - 17 \\ 15 & 12 & 10 \end{array}] + [\begin{array}{c} 8 & 4 & 4 \\ 1 & - 5 & - 9 \\ 3 & 2 & 0 \end{array}]$

Simplify $- 4 + 4$ .

$[\begin{array}{c} 0 & 0 & - 4 + 4 \\ - 1 + 1 & 5 - 5 & 9 - 9 \\ - 3 + 3 & - 2 + 2 & 0 + 0 \end{array}] - 2 X = [\begin{array}{c} - 18 & 2 & 22 \\ 9 & - 25 & - 17 \\ 15 & 12 & 10 \end{array}] + [\begin{array}{c} 8 & 4 & 4 \\ 1 & - 5 & - 9 \\ 3 & 2 & 0 \end{array}]$

Simplify $- 4 + 4$ .

$[\begin{array}{c} 0 & 0 & 0 \\ - 1 + 1 & 5 - 5 & 9 - 9 \\ - 3 + 3 & - 2 + 2 & 0 + 0 \end{array}] - 2 X = [\begin{array}{c} - 18 & 2 & 22 \\ 9 & - 25 & - 17 \\ 15 & 12 & 10 \end{array}] + [\begin{array}{c} 8 & 4 & 4 \\ 1 & - 5 & - 9 \\ 3 & 2 & 0 \end{array}]$

Simplify $- 1 + 1$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 5 - 5 & 9 - 9 \\ - 3 + 3 & - 2 + 2 & 0 + 0 \end{array}] - 2 X = [\begin{array}{c} - 18 & 2 & 22 \\ 9 & - 25 & - 17 \\ 15 & 12 & 10 \end{array}] + [\begin{array}{c} 8 & 4 & 4 \\ 1 & - 5 & - 9 \\ 3 & 2 & 0 \end{array}]$

Simplify $5 - 5$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 0 & 9 - 9 \\ - 3 + 3 & - 2 + 2 & 0 + 0 \end{array}] - 2 X = [\begin{array}{c} - 18 & 2 & 22 \\ 9 & - 25 & - 17 \\ 15 & 12 & 10 \end{array}] + [\begin{array}{c} 8 & 4 & 4 \\ 1 & - 5 & - 9 \\ 3 & 2 & 0 \end{array}]$

Simplify $9 - 9$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ - 3 + 3 & - 2 + 2 & 0 + 0 \end{array}] - 2 X = [\begin{array}{c} - 18 & 2 & 22 \\ 9 & - 25 & - 17 \\ 15 & 12 & 10 \end{array}] + [\begin{array}{c} 8 & 4 & 4 \\ 1 & - 5 & - 9 \\ 3 & 2 & 0 \end{array}]$

Simplify $- 3 + 3$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & - 2 + 2 & 0 + 0 \end{array}] - 2 X = [\begin{array}{c} - 18 & 2 & 22 \\ 9 & - 25 & - 17 \\ 15 & 12 & 10 \end{array}] + [\begin{array}{c} 8 & 4 & 4 \\ 1 & - 5 & - 9 \\ 3 & 2 & 0 \end{array}]$

Simplify $- 2 + 2$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 + 0 \end{array}] - 2 X = [\begin{array}{c} - 18 & 2 & 22 \\ 9 & - 25 & - 17 \\ 15 & 12 & 10 \end{array}] + [\begin{array}{c} 8 & 4 & 4 \\ 1 & - 5 & - 9 \\ 3 & 2 & 0 \end{array}]$

Simplify $0 + 0$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}] - 2 X = [\begin{array}{c} - 18 & 2 & 22 \\ 9 & - 25 & - 17 \\ 15 & 12 & 10 \end{array}] + [\begin{array}{c} 8 & 4 & 4 \\ 1 & - 5 & - 9 \\ 3 & 2 & 0 \end{array}]$

Simplify the right side of the equation.

Add the corresponding elements.

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}] - 2 X = [\begin{array}{c} - 18 + 8 & 2 + 4 & 22 + 4 \\ 9 + 1 & - 25 - 5 & - 17 - 9 \\ 15 + 3 & 12 + 2 & 10 + 0 \end{array}]$

Simplify each element of the matrix $[\begin{array}{c} - 18 + 8 & 2 + 4 & 22 + 4 \\ 9 + 1 & - 25 - 5 & - 17 - 9 \\ 15 + 3 & 12 + 2 & 10 + 0 \end{array}]$ .

Simplify $- 18 + 8$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}] - 2 X = [\begin{array}{c} - 10 & 2 + 4 & 22 + 4 \\ 9 + 1 & - 25 - 5 & - 17 - 9 \\ 15 + 3 & 12 + 2 & 10 + 0 \end{array}]$

Simplify $2 + 4$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}] - 2 X = [\begin{array}{c} - 10 & 6 & 22 + 4 \\ 9 + 1 & - 25 - 5 & - 17 - 9 \\ 15 + 3 & 12 + 2 & 10 + 0 \end{array}]$

Simplify $22 + 4$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}] - 2 X = [\begin{array}{c} - 10 & 6 & 26 \\ 9 + 1 & - 25 - 5 & - 17 - 9 \\ 15 + 3 & 12 + 2 & 10 + 0 \end{array}]$

Simplify $9 + 1$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}] - 2 X = [\begin{array}{c} - 10 & 6 & 26 \\ 10 & - 25 - 5 & - 17 - 9 \\ 15 + 3 & 12 + 2 & 10 + 0 \end{array}]$

Simplify $- 25 - 5$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}] - 2 X = [\begin{array}{c} - 10 & 6 & 26 \\ 10 & - 30 & - 17 - 9 \\ 15 + 3 & 12 + 2 & 10 + 0 \end{array}]$

Simplify $- 17 - 9$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}] - 2 X = [\begin{array}{c} - 10 & 6 & 26 \\ 10 & - 30 & - 26 \\ 15 + 3 & 12 + 2 & 10 + 0 \end{array}]$

Simplify $15 + 3$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}] - 2 X = [\begin{array}{c} - 10 & 6 & 26 \\ 10 & - 30 & - 26 \\ 18 & 12 + 2 & 10 + 0 \end{array}]$

Simplify $12 + 2$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}] - 2 X = [\begin{array}{c} - 10 & 6 & 26 \\ 10 & - 30 & - 26 \\ 18 & 14 & 10 + 0 \end{array}]$

Simplify $10 + 0$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}] - 2 X = [\begin{array}{c} - 10 & 6 & 26 \\ 10 & - 30 & - 26 \\ 18 & 14 & 10 \end{array}]$

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

$- 2 X = [\begin{array}{c} - 10 & 6 & 26 \\ 10 & - 30 & - 26 \\ 18 & 14 & 10 \end{array}]$

Step 3

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

$- \frac{1}{2} \cdot (- 2 X) = - \frac{1}{2} \cdot [\begin{array}{c} - 10 & 6 & 26 \\ 10 & - 30 & - 26 \\ 18 & 14 & 10 \end{array}]$

Step 4

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.

$\frac{- 1}{2} \cdot (- 2 X) = - \frac{1}{2} \cdot [\begin{array}{c} - 10 & 6 & 26 \\ 10 & - 30 & - 26 \\ 18 & 14 & 10 \end{array}]$

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

$\frac{- 1}{2} \cdot (2 (- X)) = - \frac{1}{2} \cdot [\begin{array}{c} - 10 & 6 & 26 \\ 10 & - 30 & - 26 \\ 18 & 14 & 10 \end{array}]$

Cancel the common factor.

$\frac{- 1}{2} \cdot (2 (- X)) = - \frac{1}{2} \cdot [\begin{array}{c} - 10 & 6 & 26 \\ 10 & - 30 & - 26 \\ 18 & 14 & 10 \end{array}]$

Rewrite the expression.

$- 1 \cdot (- X) = - \frac{1}{2} \cdot [\begin{array}{c} - 10 & 6 & 26 \\ 10 & - 30 & - 26 \\ 18 & 14 & 10 \end{array}]$

Multiply.

Multiply $- 1$ by $- 1$ .

$1 \cdot X = - \frac{1}{2} \cdot [\begin{array}{c} - 10 & 6 & 26 \\ 10 & - 30 & - 26 \\ 18 & 14 & 10 \end{array}]$

Multiply $X$ by $1$ .

$X = - \frac{1}{2} \cdot [\begin{array}{c} - 10 & 6 & 26 \\ 10 & - 30 & - 26 \\ 18 & 14 & 10 \end{array}]$

Simplify the right side of the equation.

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

$X = [\begin{array}{c} - \frac{1}{2} \cdot - 10 & - \frac{1}{2} \cdot 6 & - \frac{1}{2} \cdot 26 \\ - \frac{1}{2} \cdot 10 & - \frac{1}{2} \cdot - 30 & - \frac{1}{2} \cdot - 26 \\ - \frac{1}{2} \cdot 18 & - \frac{1}{2} \cdot 14 & - \frac{1}{2} \cdot 10 \end{array}]$

Simplify each element in the matrix.

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

$X = [\begin{array}{c} 5 & - \frac{1}{2} \cdot 6 & - \frac{1}{2} \cdot 26 \\ - \frac{1}{2} \cdot 10 & - \frac{1}{2} \cdot - 30 & - \frac{1}{2} \cdot - 26 \\ - \frac{1}{2} \cdot 18 & - \frac{1}{2} \cdot 14 & - \frac{1}{2} \cdot 10 \end{array}]$

Rearrange $- \frac{1}{2} \cdot 6$ .

$X = [\begin{array}{c} 5 & - 3 & - \frac{1}{2} \cdot 26 \\ - \frac{1}{2} \cdot 10 & - \frac{1}{2} \cdot - 30 & - \frac{1}{2} \cdot - 26 \\ - \frac{1}{2} \cdot 18 & - \frac{1}{2} \cdot 14 & - \frac{1}{2} \cdot 10 \end{array}]$

Rearrange $- \frac{1}{2} \cdot 26$ .

$X = [\begin{array}{c} 5 & - 3 & - 13 \\ - \frac{1}{2} \cdot 10 & - \frac{1}{2} \cdot - 30 & - \frac{1}{2} \cdot - 26 \\ - \frac{1}{2} \cdot 18 & - \frac{1}{2} \cdot 14 & - \frac{1}{2} \cdot 10 \end{array}]$

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

$X = [\begin{array}{c} 5 & - 3 & - 13 \\ - 5 & - \frac{1}{2} \cdot - 30 & - \frac{1}{2} \cdot - 26 \\ - \frac{1}{2} \cdot 18 & - \frac{1}{2} \cdot 14 & - \frac{1}{2} \cdot 10 \end{array}]$

Rearrange $- \frac{1}{2} \cdot - 30$ .

$X = [\begin{array}{c} 5 & - 3 & - 13 \\ - 5 & 15 & - \frac{1}{2} \cdot - 26 \\ - \frac{1}{2} \cdot 18 & - \frac{1}{2} \cdot 14 & - \frac{1}{2} \cdot 10 \end{array}]$

Rearrange $- \frac{1}{2} \cdot - 26$ .

$X = [\begin{array}{c} 5 & - 3 & - 13 \\ - 5 & 15 & 13 \\ - \frac{1}{2} \cdot 18 & - \frac{1}{2} \cdot 14 & - \frac{1}{2} \cdot 10 \end{array}]$

Rearrange $- \frac{1}{2} \cdot 18$ .

$X = [\begin{array}{c} 5 & - 3 & - 13 \\ - 5 & 15 & 13 \\ - 9 & - \frac{1}{2} \cdot 14 & - \frac{1}{2} \cdot 10 \end{array}]$

Rearrange $- \frac{1}{2} \cdot 14$ .

$X = [\begin{array}{c} 5 & - 3 & - 13 \\ - 5 & 15 & 13 \\ - 9 & - 7 & - \frac{1}{2} \cdot 10 \end{array}]$

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

$X = [\begin{array}{c} 5 & - 3 & - 13 \\ - 5 & 15 & 13 \\ - 9 & - 7 & - 5 \end{array}]$

Solve the Matrix Equation -2x-[[8,4,4],[1,-5,-9],[3,2,0]]=[[-18,2,22],[9,-25,-17],[15,12,10]]

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