Solve the Matrix Equation 6x+[[-20,-9,6],[-4,-22,-4],[9,-30,10]]=[[10,27,12],[20,20,14],[-9,6,-32]]

$6 x + [\begin{array}{c} - 20 & - 9 & 6 \\ - 4 & - 22 & - 4 \\ 9 & - 30 & 10 \end{array}] = [\begin{array}{c} 10 & 27 & 12 \\ 20 & 20 & 14 \\ - 9 & 6 & - 32 \end{array}]$

Step 1

Subtract the matrix from both sides of the equation.

$6 X + [\begin{array}{c} - 20 & - 9 & 6 \\ - 4 & - 22 & - 4 \\ 9 & - 30 & 10 \end{array}] - [\begin{array}{c} - 20 & - 9 & 6 \\ - 4 & - 22 & - 4 \\ 9 & - 30 & 10 \end{array}] = [\begin{array}{c} 10 & 27 & 12 \\ 20 & 20 & 14 \\ - 9 & 6 & - 32 \end{array}] - [\begin{array}{c} - 20 & - 9 & 6 \\ - 4 & - 22 & - 4 \\ 9 & - 30 & 10 \end{array}]$

Step 2

Simplify both sides of the equation.

Simplify the left side of the equation.

Subtract the corresponding elements.

$[\begin{array}{c} - 20 - (- 20) & - 9 - (- 9) & 6 - 6 \\ - 4 - (- 4) & - 22 - (- 22) & - 4 - (- 4) \\ 9 - 9 & - 30 - (- 30) & 10 - 10 \end{array}] + 6 X = [\begin{array}{c} 10 & 27 & 12 \\ 20 & 20 & 14 \\ - 9 & 6 & - 32 \end{array}] - [\begin{array}{c} - 20 & - 9 & 6 \\ - 4 & - 22 & - 4 \\ 9 & - 30 & 10 \end{array}]$

Simplify each element of the matrix $[\begin{array}{c} - 20 - (- 20) & - 9 - (- 9) & 6 - 6 \\ - 4 - (- 4) & - 22 - (- 22) & - 4 - (- 4) \\ 9 - 9 & - 30 - (- 30) & 10 - 10 \end{array}]$ .

Simplify $- 20 - (- 20)$ .

$[\begin{array}{c} 0 & - 9 - (- 9) & 6 - 6 \\ - 4 - (- 4) & - 22 - (- 22) & - 4 - (- 4) \\ 9 - 9 & - 30 - (- 30) & 10 - 10 \end{array}] + 6 X = [\begin{array}{c} 10 & 27 & 12 \\ 20 & 20 & 14 \\ - 9 & 6 & - 32 \end{array}] - [\begin{array}{c} - 20 & - 9 & 6 \\ - 4 & - 22 & - 4 \\ 9 & - 30 & 10 \end{array}]$

Simplify $- 9 - (- 9)$ .

$[\begin{array}{c} 0 & 0 & 6 - 6 \\ - 4 - (- 4) & - 22 - (- 22) & - 4 - (- 4) \\ 9 - 9 & - 30 - (- 30) & 10 - 10 \end{array}] + 6 X = [\begin{array}{c} 10 & 27 & 12 \\ 20 & 20 & 14 \\ - 9 & 6 & - 32 \end{array}] - [\begin{array}{c} - 20 & - 9 & 6 \\ - 4 & - 22 & - 4 \\ 9 & - 30 & 10 \end{array}]$

Simplify $6 - 6$ .

$[\begin{array}{c} 0 & 0 & 0 \\ - 4 - (- 4) & - 22 - (- 22) & - 4 - (- 4) \\ 9 - 9 & - 30 - (- 30) & 10 - 10 \end{array}] + 6 X = [\begin{array}{c} 10 & 27 & 12 \\ 20 & 20 & 14 \\ - 9 & 6 & - 32 \end{array}] - [\begin{array}{c} - 20 & - 9 & 6 \\ - 4 & - 22 & - 4 \\ 9 & - 30 & 10 \end{array}]$

Simplify $- 4 - (- 4)$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & - 22 - (- 22) & - 4 - (- 4) \\ 9 - 9 & - 30 - (- 30) & 10 - 10 \end{array}] + 6 X = [\begin{array}{c} 10 & 27 & 12 \\ 20 & 20 & 14 \\ - 9 & 6 & - 32 \end{array}] - [\begin{array}{c} - 20 & - 9 & 6 \\ - 4 & - 22 & - 4 \\ 9 & - 30 & 10 \end{array}]$

Simplify $- 22 - (- 22)$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 0 & - 4 - (- 4) \\ 9 - 9 & - 30 - (- 30) & 10 - 10 \end{array}] + 6 X = [\begin{array}{c} 10 & 27 & 12 \\ 20 & 20 & 14 \\ - 9 & 6 & - 32 \end{array}] - [\begin{array}{c} - 20 & - 9 & 6 \\ - 4 & - 22 & - 4 \\ 9 & - 30 & 10 \end{array}]$

Simplify $- 4 - (- 4)$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 9 - 9 & - 30 - (- 30) & 10 - 10 \end{array}] + 6 X = [\begin{array}{c} 10 & 27 & 12 \\ 20 & 20 & 14 \\ - 9 & 6 & - 32 \end{array}] - [\begin{array}{c} - 20 & - 9 & 6 \\ - 4 & - 22 & - 4 \\ 9 & - 30 & 10 \end{array}]$

Simplify $9 - 9$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & - 30 - (- 30) & 10 - 10 \end{array}] + 6 X = [\begin{array}{c} 10 & 27 & 12 \\ 20 & 20 & 14 \\ - 9 & 6 & - 32 \end{array}] - [\begin{array}{c} - 20 & - 9 & 6 \\ - 4 & - 22 & - 4 \\ 9 & - 30 & 10 \end{array}]$

Simplify $- 30 - (- 30)$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 10 - 10 \end{array}] + 6 X = [\begin{array}{c} 10 & 27 & 12 \\ 20 & 20 & 14 \\ - 9 & 6 & - 32 \end{array}] - [\begin{array}{c} - 20 & - 9 & 6 \\ - 4 & - 22 & - 4 \\ 9 & - 30 & 10 \end{array}]$

Simplify $10 - 10$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}] + 6 X = [\begin{array}{c} 10 & 27 & 12 \\ 20 & 20 & 14 \\ - 9 & 6 & - 32 \end{array}] - [\begin{array}{c} - 20 & - 9 & 6 \\ - 4 & - 22 & - 4 \\ 9 & - 30 & 10 \end{array}]$

Simplify the right side of the equation.

Subtract the corresponding elements.

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}] + 6 X = [\begin{array}{c} 10 - (- 20) & 27 - (- 9) & 12 - 6 \\ 20 - (- 4) & 20 - (- 22) & 14 - (- 4) \\ - 9 - 9 & 6 - (- 30) & - 32 - 10 \end{array}]$

Simplify each element of the matrix $[\begin{array}{c} 10 - (- 20) & 27 - (- 9) & 12 - 6 \\ 20 - (- 4) & 20 - (- 22) & 14 - (- 4) \\ - 9 - 9 & 6 - (- 30) & - 32 - 10 \end{array}]$ .

Simplify $10 - (- 20)$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}] + 6 X = [\begin{array}{c} 30 & 27 - (- 9) & 12 - 6 \\ 20 - (- 4) & 20 - (- 22) & 14 - (- 4) \\ - 9 - 9 & 6 - (- 30) & - 32 - 10 \end{array}]$

Simplify $27 - (- 9)$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}] + 6 X = [\begin{array}{c} 30 & 36 & 12 - 6 \\ 20 - (- 4) & 20 - (- 22) & 14 - (- 4) \\ - 9 - 9 & 6 - (- 30) & - 32 - 10 \end{array}]$

Simplify $12 - 6$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}] + 6 X = [\begin{array}{c} 30 & 36 & 6 \\ 20 - (- 4) & 20 - (- 22) & 14 - (- 4) \\ - 9 - 9 & 6 - (- 30) & - 32 - 10 \end{array}]$

Simplify $20 - (- 4)$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}] + 6 X = [\begin{array}{c} 30 & 36 & 6 \\ 24 & 20 - (- 22) & 14 - (- 4) \\ - 9 - 9 & 6 - (- 30) & - 32 - 10 \end{array}]$

Simplify $20 - (- 22)$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}] + 6 X = [\begin{array}{c} 30 & 36 & 6 \\ 24 & 42 & 14 - (- 4) \\ - 9 - 9 & 6 - (- 30) & - 32 - 10 \end{array}]$

Simplify $14 - (- 4)$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}] + 6 X = [\begin{array}{c} 30 & 36 & 6 \\ 24 & 42 & 18 \\ - 9 - 9 & 6 - (- 30) & - 32 - 10 \end{array}]$

Simplify $- 9 - 9$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}] + 6 X = [\begin{array}{c} 30 & 36 & 6 \\ 24 & 42 & 18 \\ - 18 & 6 - (- 30) & - 32 - 10 \end{array}]$

Simplify $6 - (- 30)$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}] + 6 X = [\begin{array}{c} 30 & 36 & 6 \\ 24 & 42 & 18 \\ - 18 & 36 & - 32 - 10 \end{array}]$

Simplify $- 32 - 10$ .

$[\begin{array}{c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}] + 6 X = [\begin{array}{c} 30 & 36 & 6 \\ 24 & 42 & 18 \\ - 18 & 36 & - 42 \end{array}]$

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

$6 X = [\begin{array}{c} 30 & 36 & 6 \\ 24 & 42 & 18 \\ - 18 & 36 & - 42 \end{array}]$

Step 3

Multiply both sides of the equation by $\frac{1}{6}$ .

$\frac{1}{6} \cdot (6 X) = \frac{1}{6} \cdot [\begin{array}{c} 30 & 36 & 6 \\ 24 & 42 & 18 \\ - 18 & 36 & - 42 \end{array}]$

Step 4

Simplify both sides of the equation.

Cancel the common factor of $6$ .

Factor $6$ out of $6 X$ .

$\frac{1}{6} \cdot (6 (X)) = \frac{1}{6} \cdot [\begin{array}{c} 30 & 36 & 6 \\ 24 & 42 & 18 \\ - 18 & 36 & - 42 \end{array}]$

Cancel the common factor.

$\frac{1}{6} \cdot (6 X) = \frac{1}{6} \cdot [\begin{array}{c} 30 & 36 & 6 \\ 24 & 42 & 18 \\ - 18 & 36 & - 42 \end{array}]$

Rewrite the expression.

$X = \frac{1}{6} \cdot [\begin{array}{c} 30 & 36 & 6 \\ 24 & 42 & 18 \\ - 18 & 36 & - 42 \end{array}]$

Simplify the right side of the equation.

Multiply $\frac{1}{6}$ by each element of the matrix.

$X = [\begin{array}{c} \frac{1}{6} \cdot 30 & \frac{1}{6} \cdot 36 & \frac{1}{6} \cdot 6 \\ \frac{1}{6} \cdot 24 & \frac{1}{6} \cdot 42 & \frac{1}{6} \cdot 18 \\ \frac{1}{6} \cdot - 18 & \frac{1}{6} \cdot 36 & \frac{1}{6} \cdot - 42 \end{array}]$

Simplify each element in the matrix.

Rearrange $\frac{1}{6} \cdot 30$ .

$X = [\begin{array}{c} 5 & \frac{1}{6} \cdot 36 & \frac{1}{6} \cdot 6 \\ \frac{1}{6} \cdot 24 & \frac{1}{6} \cdot 42 & \frac{1}{6} \cdot 18 \\ \frac{1}{6} \cdot - 18 & \frac{1}{6} \cdot 36 & \frac{1}{6} \cdot - 42 \end{array}]$

Rearrange $\frac{1}{6} \cdot 36$ .

$X = [\begin{array}{c} 5 & 6 & \frac{1}{6} \cdot 6 \\ \frac{1}{6} \cdot 24 & \frac{1}{6} \cdot 42 & \frac{1}{6} \cdot 18 \\ \frac{1}{6} \cdot - 18 & \frac{1}{6} \cdot 36 & \frac{1}{6} \cdot - 42 \end{array}]$

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

$X = [\begin{array}{c} 5 & 6 & 1 \\ \frac{1}{6} \cdot 24 & \frac{1}{6} \cdot 42 & \frac{1}{6} \cdot 18 \\ \frac{1}{6} \cdot - 18 & \frac{1}{6} \cdot 36 & \frac{1}{6} \cdot - 42 \end{array}]$

Rearrange $\frac{1}{6} \cdot 24$ .

$X = [\begin{array}{c} 5 & 6 & 1 \\ 4 & \frac{1}{6} \cdot 42 & \frac{1}{6} \cdot 18 \\ \frac{1}{6} \cdot - 18 & \frac{1}{6} \cdot 36 & \frac{1}{6} \cdot - 42 \end{array}]$

Rearrange $\frac{1}{6} \cdot 42$ .

$X = [\begin{array}{c} 5 & 6 & 1 \\ 4 & 7 & \frac{1}{6} \cdot 18 \\ \frac{1}{6} \cdot - 18 & \frac{1}{6} \cdot 36 & \frac{1}{6} \cdot - 42 \end{array}]$

Rearrange $\frac{1}{6} \cdot 18$ .

$X = [\begin{array}{c} 5 & 6 & 1 \\ 4 & 7 & 3 \\ \frac{1}{6} \cdot - 18 & \frac{1}{6} \cdot 36 & \frac{1}{6} \cdot - 42 \end{array}]$

Rearrange $\frac{1}{6} \cdot - 18$ .

$X = [\begin{array}{c} 5 & 6 & 1 \\ 4 & 7 & 3 \\ - 3 & \frac{1}{6} \cdot 36 & \frac{1}{6} \cdot - 42 \end{array}]$

Rearrange $\frac{1}{6} \cdot 36$ .

$X = [\begin{array}{c} 5 & 6 & 1 \\ 4 & 7 & 3 \\ - 3 & 6 & \frac{1}{6} \cdot - 42 \end{array}]$

Rearrange $\frac{1}{6} \cdot - 42$ .

$X = [\begin{array}{c} 5 & 6 & 1 \\ 4 & 7 & 3 \\ - 3 & 6 & - 7 \end{array}]$

Solve the Matrix Equation 6x+[[-20,-9,6],[-4,-22,-4],[9,-30,10]]=[[10,27,12],[20,20,14],[-9,6,-32]]

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