Solve the Matrix Equation -1/3X-[[3,6,2],[9,1,7]]=[[-8,3,-1],[12,-4,-3]]

$- \frac{1}{3} X - [\begin{array}{c} 3 & 6 & 2 \\ 9 & 1 & 7 \end{array}] = [\begin{array}{c} - 8 & 3 & - 1 \\ 12 & - 4 & - 3 \end{array}]$

Add the matrix to both sides of the equation.

$- \frac{1}{3} X - [\begin{array}{c} 3 & 6 & 2 \\ 9 & 1 & 7 \end{array}] + [\begin{array}{c} 3 & 6 & 2 \\ 9 & 1 & 7 \end{array}] = [\begin{array}{c} - 8 & 3 & - 1 \\ 12 & - 4 & - 3 \end{array}] + [\begin{array}{c} 3 & 6 & 2 \\ 9 & 1 & 7 \end{array}]$

Simplify both sides of the equation.

Simplify the left side of the equation.

Combine $X$ and $\frac{1}{3}$ .

$- [\begin{array}{c} 3 & 6 & 2 \\ 9 & 1 & 7 \end{array}] + [\begin{array}{c} 3 & 6 & 2 \\ 9 & 1 & 7 \end{array}] - \frac{X}{3} = [\begin{array}{c} - 8 & 3 & - 1 \\ 12 & - 4 & - 3 \end{array}] + [\begin{array}{c} 3 & 6 & 2 \\ 9 & 1 & 7 \end{array}]$

Add the corresponding elements.

$[\begin{array}{c} - 3 + 3 & - 6 + 6 & - 2 + 2 \\ - 9 + 9 & - 1 + 1 & - 7 + 7 \end{array}] - \frac{X}{3} = [\begin{array}{c} - 8 & 3 & - 1 \\ 12 & - 4 & - 3 \end{array}] + [\begin{array}{c} 3 & 6 & 2 \\ 9 & 1 & 7 \end{array}]$

Simplify each element of the matrix $[\begin{array}{c} - 3 + 3 & - 6 + 6 & - 2 + 2 \\ - 9 + 9 & - 1 + 1 & - 7 + 7 \end{array}]$ .

Simplify $- 3 + 3$ .

$[\begin{array}{c} 0 & - 6 + 6 & - 2 + 2 \\ - 9 + 9 & - 1 + 1 & - 7 + 7 \end{array}] - \frac{X}{3} = [\begin{array}{c} - 8 & 3 & - 1 \\ 12 & - 4 & - 3 \end{array}] + [\begin{array}{c} 3 & 6 & 2 \\ 9 & 1 & 7 \end{array}]$

Simplify $- 6 + 6$ .

$[\begin{array}{c} 0 & 0 & - 2 + 2 \\ - 9 + 9 & - 1 + 1 & - 7 + 7 \end{array}] - \frac{X}{3} = [\begin{array}{c} - 8 & 3 & - 1 \\ 12 & - 4 & - 3 \end{array}] + [\begin{array}{c} 3 & 6 & 2 \\ 9 & 1 & 7 \end{array}]$

Simplify $- 2 + 2$ .

$[\begin{array}{c} 0 & 0 & 0 \\ - 9 + 9 & - 1 + 1 & - 7 + 7 \end{array}] - \frac{X}{3} = [\begin{array}{c} - 8 & 3 & - 1 \\ 12 & - 4 & - 3 \end{array}] + [\begin{array}{c} 3 & 6 & 2 \\ 9 & 1 & 7 \end{array}]$

Simplify $- 9 + 9$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & - 1 + 1 & - 7 + 7 \end{array}] - \frac{X}{3} = [\begin{array}{c} - 8 & 3 & - 1 \\ 12 & - 4 & - 3 \end{array}] + [\begin{array}{c} 3 & 6 & 2 \\ 9 & 1 & 7 \end{array}]$

Simplify $- 1 + 1$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 0 & - 7 + 7 \end{array}] - \frac{X}{3} = [\begin{array}{c} - 8 & 3 & - 1 \\ 12 & - 4 & - 3 \end{array}] + [\begin{array}{c} 3 & 6 & 2 \\ 9 & 1 & 7 \end{array}]$

Simplify $- 7 + 7$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}] - \frac{X}{3} = [\begin{array}{c} - 8 & 3 & - 1 \\ 12 & - 4 & - 3 \end{array}] + [\begin{array}{c} 3 & 6 & 2 \\ 9 & 1 & 7 \end{array}]$

Simplify the right side of the equation.

Add the corresponding elements.

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}] - \frac{X}{3} = [\begin{array}{c} - 8 + 3 & 3 + 6 & - 1 + 2 \\ 12 + 9 & - 4 + 1 & - 3 + 7 \end{array}]$

Simplify each element of the matrix $[\begin{array}{c} - 8 + 3 & 3 + 6 & - 1 + 2 \\ 12 + 9 & - 4 + 1 & - 3 + 7 \end{array}]$ .

Simplify $- 8 + 3$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}] - \frac{X}{3} = [\begin{array}{c} - 5 & 3 + 6 & - 1 + 2 \\ 12 + 9 & - 4 + 1 & - 3 + 7 \end{array}]$

Simplify $3 + 6$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}] - \frac{X}{3} = [\begin{array}{c} - 5 & 9 & - 1 + 2 \\ 12 + 9 & - 4 + 1 & - 3 + 7 \end{array}]$

Simplify $- 1 + 2$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}] - \frac{X}{3} = [\begin{array}{c} - 5 & 9 & 1 \\ 12 + 9 & - 4 + 1 & - 3 + 7 \end{array}]$

Simplify $12 + 9$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}] - \frac{X}{3} = [\begin{array}{c} - 5 & 9 & 1 \\ 21 & - 4 + 1 & - 3 + 7 \end{array}]$

Simplify $- 4 + 1$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}] - \frac{X}{3} = [\begin{array}{c} - 5 & 9 & 1 \\ 21 & - 3 & - 3 + 7 \end{array}]$

Simplify $- 3 + 7$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}] - \frac{X}{3} = [\begin{array}{c} - 5 & 9 & 1 \\ 21 & - 3 & 4 \end{array}]$

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

$- \frac{X}{3} = [\begin{array}{c} - 5 & 9 & 1 \\ 21 & - 3 & 4 \end{array}]$

Multiply both sides of the equation by $- 3$ .

$- 3 \cdot (- \frac{X}{3}) = - 3 \cdot [\begin{array}{c} - 5 & 9 & 1 \\ 21 & - 3 & 4 \end{array}]$

Simplify both sides of the equation.

Simplify the left side of the equation.

Cancel the common factor of $3$ .

Move the leading negative in $- \frac{X}{3}$ into the numerator.

$- 3 \cdot \frac{- X}{3} = - 3 \cdot [\begin{array}{c} - 5 & 9 & 1 \\ 21 & - 3 & 4 \end{array}]$

Factor $3$ out of $- 3$ .

$3 (- 1) \cdot \frac{- X}{3} = - 3 \cdot [\begin{array}{c} - 5 & 9 & 1 \\ 21 & - 3 & 4 \end{array}]$

Cancel the common factor.

$3 \cdot - 1 \cdot \frac{- X}{3} = - 3 \cdot [\begin{array}{c} - 5 & 9 & 1 \\ 21 & - 3 & 4 \end{array}]$

Rewrite the expression.

$- 1 \cdot (- X) = - 3 \cdot [\begin{array}{c} - 5 & 9 & 1 \\ 21 & - 3 & 4 \end{array}]$

Multiply.

Multiply $- 1$ by $- 1$ .

$1 \cdot X = - 3 \cdot [\begin{array}{c} - 5 & 9 & 1 \\ 21 & - 3 & 4 \end{array}]$

Multiply $X$ by $1$ .

$X = - 3 \cdot [\begin{array}{c} - 5 & 9 & 1 \\ 21 & - 3 & 4 \end{array}]$

Simplify the right side of the equation.

Multiply $- 3$ by each element of the matrix.

$X = [\begin{array}{c} - 3 \cdot - 5 & - 3 \cdot 9 & - 3 \cdot 1 \\ - 3 \cdot 21 & - 3 \cdot - 3 & - 3 \cdot 4 \end{array}]$

Simplify each element in the matrix.

Rearrange $- 3 \cdot - 5$ .

$X = [\begin{array}{c} 15 & - 3 \cdot 9 & - 3 \cdot 1 \\ - 3 \cdot 21 & - 3 \cdot - 3 & - 3 \cdot 4 \end{array}]$

Rearrange $- 3 \cdot 9$ .

$X = [\begin{array}{c} 15 & - 27 & - 3 \cdot 1 \\ - 3 \cdot 21 & - 3 \cdot - 3 & - 3 \cdot 4 \end{array}]$

Rearrange $- 3 \cdot 1$ .

$X = [\begin{array}{c} 15 & - 27 & - 3 \\ - 3 \cdot 21 & - 3 \cdot - 3 & - 3 \cdot 4 \end{array}]$

Rearrange $- 3 \cdot 21$ .

$X = [\begin{array}{c} 15 & - 27 & - 3 \\ - 63 & - 3 \cdot - 3 & - 3 \cdot 4 \end{array}]$

Rearrange $- 3 \cdot - 3$ .

$X = [\begin{array}{c} 15 & - 27 & - 3 \\ - 63 & 9 & - 3 \cdot 4 \end{array}]$

Rearrange $- 3 \cdot 4$ .

$X = [\begin{array}{c} 15 & - 27 & - 3 \\ - 63 & 9 & - 12 \end{array}]$

Solve the Matrix Equation -1/3X-[[3,6,2],[9,1,7]]=[[-8,3,-1],[12,-4,-3]]

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