Solve the Matrix Equation 2x-[[-4,6],[2,-3]]=[[2,-2],[8,5]]

$2 x - [\begin{array}{c} - 4 & 6 \\ 2 & - 3 \end{array}] = [\begin{array}{c} 2 & - 2 \\ 8 & 5 \end{array}]$

Step 1

Add the matrix to both sides of the equation.

$2 X - [\begin{array}{c} - 4 & 6 \\ 2 & - 3 \end{array}] + [\begin{array}{c} - 4 & 6 \\ 2 & - 3 \end{array}] = [\begin{array}{c} 2 & - 2 \\ 8 & 5 \end{array}] + [\begin{array}{c} - 4 & 6 \\ 2 & - 3 \end{array}]$

Step 2

Simplify both sides of the equation.

Simplify the left side of the equation.

Add the corresponding elements.

$[\begin{array}{c} 4 - 4 & - 6 + 6 \\ - 2 + 2 & 3 - 3 \end{array}] + 2 X = [\begin{array}{c} 2 & - 2 \\ 8 & 5 \end{array}] + [\begin{array}{c} - 4 & 6 \\ 2 & - 3 \end{array}]$

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

Simplify $4 - 4$ .

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

Simplify $- 6 + 6$ .

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

Simplify $- 2 + 2$ .

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

Simplify $3 - 3$ .

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

Simplify the right side of the equation.

Add the corresponding elements.

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

Simplify each element of the matrix $[\begin{array}{c} 2 - 4 & - 2 + 6 \\ 8 + 2 & 5 - 3 \end{array}]$ .

Simplify $2 - 4$ .

$[\begin{array}{c} 0 & 0 \\ 0 & 0 \end{array}] + 2 X = [\begin{array}{c} - 2 & - 2 + 6 \\ 8 + 2 & 5 - 3 \end{array}]$

Simplify $- 2 + 6$ .

$[\begin{array}{c} 0 & 0 \\ 0 & 0 \end{array}] + 2 X = [\begin{array}{c} - 2 & 4 \\ 8 + 2 & 5 - 3 \end{array}]$

Simplify $8 + 2$ .

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

Simplify $5 - 3$ .

$[\begin{array}{c} 0 & 0 \\ 0 & 0 \end{array}] + 2 X = [\begin{array}{c} - 2 & 4 \\ 10 & 2 \end{array}]$

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

$2 X = [\begin{array}{c} - 2 & 4 \\ 10 & 2 \end{array}]$

Step 3

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

$\frac{1}{2} \cdot (2 X) = \frac{1}{2} \cdot [\begin{array}{c} - 2 & 4 \\ 10 & 2 \end{array}]$

Step 4

Simplify both sides of the equation.

Cancel the common factor of $2$ .

Factor $2$ out of $2 X$ .

$\frac{1}{2} \cdot (2 (X)) = \frac{1}{2} \cdot [\begin{array}{c} - 2 & 4 \\ 10 & 2 \end{array}]$

Cancel the common factor.

$\frac{1}{2} \cdot (2 X) = \frac{1}{2} \cdot [\begin{array}{c} - 2 & 4 \\ 10 & 2 \end{array}]$

Rewrite the expression.

$X = \frac{1}{2} \cdot [\begin{array}{c} - 2 & 4 \\ 10 & 2 \end{array}]$

Simplify the right side of the equation.

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

$X = [\begin{array}{c} \frac{1}{2} \cdot - 2 & \frac{1}{2} \cdot 4 \\ \frac{1}{2} \cdot 10 & \frac{1}{2} \cdot 2 \end{array}]$

Simplify each element in the matrix.

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

$X = [\begin{array}{c} - 1 & \frac{1}{2} \cdot 4 \\ \frac{1}{2} \cdot 10 & \frac{1}{2} \cdot 2 \end{array}]$

Rearrange $\frac{1}{2} \cdot 4$ .

$X = [\begin{array}{c} - 1 & 2 \\ \frac{1}{2} \cdot 10 & \frac{1}{2} \cdot 2 \end{array}]$

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

$X = [\begin{array}{c} - 1 & 2 \\ 5 & \frac{1}{2} \cdot 2 \end{array}]$

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

$X = [\begin{array}{c} - 1 & 2 \\ 5 & 1 \end{array}]$

Solve the Matrix Equation 2x-[[-4,6],[2,-3]]=[[2,-2],[8,5]]

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