Solve the Matrix Equation 1/3x-[[16,4,2],[8,1,6],[9,-2,1]]=[[-4,-8,3],[-2,12,10],[-9,3,1]]

$\frac{1}{3} x - [\begin{array}{c} 16 & 4 & 2 \\ 8 & 1 & 6 \\ 9 & - 2 & 1 \end{array}] = [\begin{array}{c} - 4 & - 8 & 3 \\ - 2 & 12 & 10 \\ - 9 & 3 & 1 \end{array}]$

Step 1

Add the matrix to both sides of the equation.

$\frac{1}{3} X - [\begin{array}{c} 16 & 4 & 2 \\ 8 & 1 & 6 \\ 9 & - 2 & 1 \end{array}] + [\begin{array}{c} 16 & 4 & 2 \\ 8 & 1 & 6 \\ 9 & - 2 & 1 \end{array}] = [\begin{array}{c} - 4 & - 8 & 3 \\ - 2 & 12 & 10 \\ - 9 & 3 & 1 \end{array}] + [\begin{array}{c} 16 & 4 & 2 \\ 8 & 1 & 6 \\ 9 & - 2 & 1 \end{array}]$

Step 2

Simplify both sides of the equation.

Simplify the left side of the equation.

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

$- [\begin{array}{c} 16 & 4 & 2 \\ 8 & 1 & 6 \\ 9 & - 2 & 1 \end{array}] + [\begin{array}{c} 16 & 4 & 2 \\ 8 & 1 & 6 \\ 9 & - 2 & 1 \end{array}] + \frac{X}{3} = [\begin{array}{c} - 4 & - 8 & 3 \\ - 2 & 12 & 10 \\ - 9 & 3 & 1 \end{array}] + [\begin{array}{c} 16 & 4 & 2 \\ 8 & 1 & 6 \\ 9 & - 2 & 1 \end{array}]$

Add the corresponding elements.

$[\begin{array}{c} - 16 + 16 & - 4 + 4 & - 2 + 2 \\ - 8 + 8 & - 1 + 1 & - 6 + 6 \\ - 9 + 9 & 2 - 2 & - 1 + 1 \end{array}] + \frac{X}{3} = [\begin{array}{c} - 4 & - 8 & 3 \\ - 2 & 12 & 10 \\ - 9 & 3 & 1 \end{array}] + [\begin{array}{c} 16 & 4 & 2 \\ 8 & 1 & 6 \\ 9 & - 2 & 1 \end{array}]$

Simplify each element of the matrix $[\begin{array}{c} - 16 + 16 & - 4 + 4 & - 2 + 2 \\ - 8 + 8 & - 1 + 1 & - 6 + 6 \\ - 9 + 9 & 2 - 2 & - 1 + 1 \end{array}]$ .

Simplify $- 16 + 16$ .

$[\begin{array}{c} 0 & - 4 + 4 & - 2 + 2 \\ - 8 + 8 & - 1 + 1 & - 6 + 6 \\ - 9 + 9 & 2 - 2 & - 1 + 1 \end{array}] + \frac{X}{3} = [\begin{array}{c} - 4 & - 8 & 3 \\ - 2 & 12 & 10 \\ - 9 & 3 & 1 \end{array}] + [\begin{array}{c} 16 & 4 & 2 \\ 8 & 1 & 6 \\ 9 & - 2 & 1 \end{array}]$

Simplify $- 4 + 4$ .

$[\begin{array}{c} 0 & 0 & - 2 + 2 \\ - 8 + 8 & - 1 + 1 & - 6 + 6 \\ - 9 + 9 & 2 - 2 & - 1 + 1 \end{array}] + \frac{X}{3} = [\begin{array}{c} - 4 & - 8 & 3 \\ - 2 & 12 & 10 \\ - 9 & 3 & 1 \end{array}] + [\begin{array}{c} 16 & 4 & 2 \\ 8 & 1 & 6 \\ 9 & - 2 & 1 \end{array}]$

Simplify $- 2 + 2$ .

$[\begin{array}{c} 0 & 0 & 0 \\ - 8 + 8 & - 1 + 1 & - 6 + 6 \\ - 9 + 9 & 2 - 2 & - 1 + 1 \end{array}] + \frac{X}{3} = [\begin{array}{c} - 4 & - 8 & 3 \\ - 2 & 12 & 10 \\ - 9 & 3 & 1 \end{array}] + [\begin{array}{c} 16 & 4 & 2 \\ 8 & 1 & 6 \\ 9 & - 2 & 1 \end{array}]$

Simplify $- 8 + 8$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & - 1 + 1 & - 6 + 6 \\ - 9 + 9 & 2 - 2 & - 1 + 1 \end{array}] + \frac{X}{3} = [\begin{array}{c} - 4 & - 8 & 3 \\ - 2 & 12 & 10 \\ - 9 & 3 & 1 \end{array}] + [\begin{array}{c} 16 & 4 & 2 \\ 8 & 1 & 6 \\ 9 & - 2 & 1 \end{array}]$

Simplify $- 1 + 1$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 0 & - 6 + 6 \\ - 9 + 9 & 2 - 2 & - 1 + 1 \end{array}] + \frac{X}{3} = [\begin{array}{c} - 4 & - 8 & 3 \\ - 2 & 12 & 10 \\ - 9 & 3 & 1 \end{array}] + [\begin{array}{c} 16 & 4 & 2 \\ 8 & 1 & 6 \\ 9 & - 2 & 1 \end{array}]$

Simplify $- 6 + 6$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ - 9 + 9 & 2 - 2 & - 1 + 1 \end{array}] + \frac{X}{3} = [\begin{array}{c} - 4 & - 8 & 3 \\ - 2 & 12 & 10 \\ - 9 & 3 & 1 \end{array}] + [\begin{array}{c} 16 & 4 & 2 \\ 8 & 1 & 6 \\ 9 & - 2 & 1 \end{array}]$

Simplify $- 9 + 9$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 2 - 2 & - 1 + 1 \end{array}] + \frac{X}{3} = [\begin{array}{c} - 4 & - 8 & 3 \\ - 2 & 12 & 10 \\ - 9 & 3 & 1 \end{array}] + [\begin{array}{c} 16 & 4 & 2 \\ 8 & 1 & 6 \\ 9 & - 2 & 1 \end{array}]$

Simplify $2 - 2$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & - 1 + 1 \end{array}] + \frac{X}{3} = [\begin{array}{c} - 4 & - 8 & 3 \\ - 2 & 12 & 10 \\ - 9 & 3 & 1 \end{array}] + [\begin{array}{c} 16 & 4 & 2 \\ 8 & 1 & 6 \\ 9 & - 2 & 1 \end{array}]$

Simplify $- 1 + 1$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}] + \frac{X}{3} = [\begin{array}{c} - 4 & - 8 & 3 \\ - 2 & 12 & 10 \\ - 9 & 3 & 1 \end{array}] + [\begin{array}{c} 16 & 4 & 2 \\ 8 & 1 & 6 \\ 9 & - 2 & 1 \end{array}]$

Simplify the right side of the equation.

Add the corresponding elements.

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}] + \frac{X}{3} = [\begin{array}{c} - 4 + 16 & - 8 + 4 & 3 + 2 \\ - 2 + 8 & 12 + 1 & 10 + 6 \\ - 9 + 9 & 3 - 2 & 1 + 1 \end{array}]$

Simplify each element of the matrix $[\begin{array}{c} - 4 + 16 & - 8 + 4 & 3 + 2 \\ - 2 + 8 & 12 + 1 & 10 + 6 \\ - 9 + 9 & 3 - 2 & 1 + 1 \end{array}]$ .

Simplify $- 4 + 16$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}] + \frac{X}{3} = [\begin{array}{c} 12 & - 8 + 4 & 3 + 2 \\ - 2 + 8 & 12 + 1 & 10 + 6 \\ - 9 + 9 & 3 - 2 & 1 + 1 \end{array}]$

Simplify $- 8 + 4$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}] + \frac{X}{3} = [\begin{array}{c} 12 & - 4 & 3 + 2 \\ - 2 + 8 & 12 + 1 & 10 + 6 \\ - 9 + 9 & 3 - 2 & 1 + 1 \end{array}]$

Simplify $3 + 2$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}] + \frac{X}{3} = [\begin{array}{c} 12 & - 4 & 5 \\ - 2 + 8 & 12 + 1 & 10 + 6 \\ - 9 + 9 & 3 - 2 & 1 + 1 \end{array}]$

Simplify $- 2 + 8$ .

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

Simplify $12 + 1$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}] + \frac{X}{3} = [\begin{array}{c} 12 & - 4 & 5 \\ 6 & 13 & 10 + 6 \\ - 9 + 9 & 3 - 2 & 1 + 1 \end{array}]$

Simplify $10 + 6$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}] + \frac{X}{3} = [\begin{array}{c} 12 & - 4 & 5 \\ 6 & 13 & 16 \\ - 9 + 9 & 3 - 2 & 1 + 1 \end{array}]$

Simplify $- 9 + 9$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}] + \frac{X}{3} = [\begin{array}{c} 12 & - 4 & 5 \\ 6 & 13 & 16 \\ 0 & 3 - 2 & 1 + 1 \end{array}]$

Simplify $3 - 2$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}] + \frac{X}{3} = [\begin{array}{c} 12 & - 4 & 5 \\ 6 & 13 & 16 \\ 0 & 1 & 1 + 1 \end{array}]$

Simplify $1 + 1$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}] + \frac{X}{3} = [\begin{array}{c} 12 & - 4 & 5 \\ 6 & 13 & 16 \\ 0 & 1 & 2 \end{array}]$

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

$\frac{X}{3} = [\begin{array}{c} 12 & - 4 & 5 \\ 6 & 13 & 16 \\ 0 & 1 & 2 \end{array}]$

Step 3

Multiply both sides of the equation by $3$ .

$3 \cdot \frac{X}{3} = 3 \cdot [\begin{array}{c} 12 & - 4 & 5 \\ 6 & 13 & 16 \\ 0 & 1 & 2 \end{array}]$

Step 4

Simplify both sides of the equation.

Cancel the common factor of $3$ .

Cancel the common factor.

$3 \cdot \frac{X}{3} = 3 \cdot [\begin{array}{c} 12 & - 4 & 5 \\ 6 & 13 & 16 \\ 0 & 1 & 2 \end{array}]$

Rewrite the expression.

$X = 3 \cdot [\begin{array}{c} 12 & - 4 & 5 \\ 6 & 13 & 16 \\ 0 & 1 & 2 \end{array}]$

Simplify the right side of the equation.

Multiply $3$ by each element of the matrix.

$X = [\begin{array}{c} 3 \cdot 12 & 3 \cdot - 4 & 3 \cdot 5 \\ 3 \cdot 6 & 3 \cdot 13 & 3 \cdot 16 \\ 3 \cdot 0 & 3 \cdot 1 & 3 \cdot 2 \end{array}]$

Simplify each element in the matrix.

Rearrange $3 \cdot 12$ .

$X = [\begin{array}{c} 36 & 3 \cdot - 4 & 3 \cdot 5 \\ 3 \cdot 6 & 3 \cdot 13 & 3 \cdot 16 \\ 3 \cdot 0 & 3 \cdot 1 & 3 \cdot 2 \end{array}]$

Rearrange $3 \cdot - 4$ .

$X = [\begin{array}{c} 36 & - 12 & 3 \cdot 5 \\ 3 \cdot 6 & 3 \cdot 13 & 3 \cdot 16 \\ 3 \cdot 0 & 3 \cdot 1 & 3 \cdot 2 \end{array}]$

Rearrange $3 \cdot 5$ .

$X = [\begin{array}{c} 36 & - 12 & 15 \\ 3 \cdot 6 & 3 \cdot 13 & 3 \cdot 16 \\ 3 \cdot 0 & 3 \cdot 1 & 3 \cdot 2 \end{array}]$

Rearrange $3 \cdot 6$ .

$X = [\begin{array}{c} 36 & - 12 & 15 \\ 18 & 3 \cdot 13 & 3 \cdot 16 \\ 3 \cdot 0 & 3 \cdot 1 & 3 \cdot 2 \end{array}]$

Rearrange $3 \cdot 13$ .

$X = [\begin{array}{c} 36 & - 12 & 15 \\ 18 & 39 & 3 \cdot 16 \\ 3 \cdot 0 & 3 \cdot 1 & 3 \cdot 2 \end{array}]$

Rearrange $3 \cdot 16$ .

$X = [\begin{array}{c} 36 & - 12 & 15 \\ 18 & 39 & 48 \\ 3 \cdot 0 & 3 \cdot 1 & 3 \cdot 2 \end{array}]$

Rearrange $3 \cdot 0$ .

$X = [\begin{array}{c} 36 & - 12 & 15 \\ 18 & 39 & 48 \\ 0 & 3 \cdot 1 & 3 \cdot 2 \end{array}]$

Rearrange $3 \cdot 1$ .

$X = [\begin{array}{c} 36 & - 12 & 15 \\ 18 & 39 & 48 \\ 0 & 3 & 3 \cdot 2 \end{array}]$

Rearrange $3 \cdot 2$ .

$X = [\begin{array}{c} 36 & - 12 & 15 \\ 18 & 39 & 48 \\ 0 & 3 & 6 \end{array}]$

Solve the Matrix Equation 1/3x-[[16,4,2],[8,1,6],[9,-2,1]]=[[-4,-8,3],[-2,12,10],[-9,3,1]]

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