Solve the Matrix Equation [[4,-3,2],[0,5,-8]]-2x=[[2,5,4],[4,5,2]]

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

Step 1

Move all terms not containing a variable to the right side.

Subtract $[\begin{array}{c} 4 & - 3 & 2 \\ 0 & 5 & - 8 \end{array}]$ from both sides of the equation.

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

Step 2

Simplify the right side.

Subtract the corresponding elements.

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

Simplify each element.

Subtract $4$ from $2$ .

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

Add $5$ and $3$ .

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

Subtract $2$ from $4$ .

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

Subtract $0$ from $4$ .

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

Subtract $5$ from $5$ .

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

Add $2$ and $8$ .

$- 2 x = [\begin{array}{c} - 2 & 8 & 2 \\ 4 & 0 & 10 \end{array}]$

Step 3

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

$- \frac{1}{2} (- 2 x) = - \frac{1}{2} [\begin{array}{c} - 2 & 8 & 2 \\ 4 & 0 & 10 \end{array}]$

Step 4

Simplify.

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{1}{2}$ into the numerator.

$\frac{- 1}{2} (- 2 x) = - \frac{1}{2} [\begin{array}{c} - 2 & 8 & 2 \\ 4 & 0 & 10 \end{array}]$

Factor $2$ out of $- 2 x$ .

$\frac{- 1}{2} (2 (- x)) = - \frac{1}{2} [\begin{array}{c} - 2 & 8 & 2 \\ 4 & 0 & 10 \end{array}]$

Cancel the common factor.

$\frac{- 1}{2} (2 (- x)) = - \frac{1}{2} [\begin{array}{c} - 2 & 8 & 2 \\ 4 & 0 & 10 \end{array}]$

Rewrite the expression.

$- - x = - \frac{1}{2} [\begin{array}{c} - 2 & 8 & 2 \\ 4 & 0 & 10 \end{array}]$

Multiply $- 1$ by $- 1$ .

$1 x = - \frac{1}{2} [\begin{array}{c} - 2 & 8 & 2 \\ 4 & 0 & 10 \end{array}]$

Multiply $x$ by $1$ .

$x = - \frac{1}{2} [\begin{array}{c} - 2 & 8 & 2 \\ 4 & 0 & 10 \end{array}]$

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

$x = [\begin{array}{c} - \frac{1}{2} \cdot - 2 & - \frac{1}{2} \cdot 8 & - \frac{1}{2} \cdot 2 \\ - \frac{1}{2} \cdot 4 & - \frac{1}{2} \cdot 0 & - \frac{1}{2} \cdot 10 \end{array}]$

Simplify each element in the matrix.

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{1}{2}$ into the numerator.

$x = [\begin{array}{c} \frac{- 1}{2} \cdot - 2 & - \frac{1}{2} \cdot 8 & - \frac{1}{2} \cdot 2 \\ - \frac{1}{2} \cdot 4 & - \frac{1}{2} \cdot 0 & - \frac{1}{2} \cdot 10 \end{array}]$

Factor $2$ out of $- 2$ .

$x = [\begin{array}{c} \frac{- 1}{2} \cdot (2 (- 1)) & - \frac{1}{2} \cdot 8 & - \frac{1}{2} \cdot 2 \\ - \frac{1}{2} \cdot 4 & - \frac{1}{2} \cdot 0 & - \frac{1}{2} \cdot 10 \end{array}]$

Cancel the common factor.

$x = [\begin{array}{c} \frac{- 1}{2} \cdot (2 \cdot - 1) & - \frac{1}{2} \cdot 8 & - \frac{1}{2} \cdot 2 \\ - \frac{1}{2} \cdot 4 & - \frac{1}{2} \cdot 0 & - \frac{1}{2} \cdot 10 \end{array}]$

Rewrite the expression.

$x = [\begin{array}{c} - 1 \cdot - 1 & - \frac{1}{2} \cdot 8 & - \frac{1}{2} \cdot 2 \\ - \frac{1}{2} \cdot 4 & - \frac{1}{2} \cdot 0 & - \frac{1}{2} \cdot 10 \end{array}]$

Multiply $- 1$ by $- 1$ .

$x = [\begin{array}{c} 1 & - \frac{1}{2} \cdot 8 & - \frac{1}{2} \cdot 2 \\ - \frac{1}{2} \cdot 4 & - \frac{1}{2} \cdot 0 & - \frac{1}{2} \cdot 10 \end{array}]$

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{1}{2}$ into the numerator.

$x = [\begin{array}{c} 1 & \frac{- 1}{2} \cdot 8 & - \frac{1}{2} \cdot 2 \\ - \frac{1}{2} \cdot 4 & - \frac{1}{2} \cdot 0 & - \frac{1}{2} \cdot 10 \end{array}]$

Factor $2$ out of $8$ .

$x = [\begin{array}{c} 1 & \frac{- 1}{2} \cdot (2 (4)) & - \frac{1}{2} \cdot 2 \\ - \frac{1}{2} \cdot 4 & - \frac{1}{2} \cdot 0 & - \frac{1}{2} \cdot 10 \end{array}]$

Cancel the common factor.

$x = [\begin{array}{c} 1 & \frac{- 1}{2} \cdot (2 \cdot 4) & - \frac{1}{2} \cdot 2 \\ - \frac{1}{2} \cdot 4 & - \frac{1}{2} \cdot 0 & - \frac{1}{2} \cdot 10 \end{array}]$

Rewrite the expression.

$x = [\begin{array}{c} 1 & - 1 \cdot 4 & - \frac{1}{2} \cdot 2 \\ - \frac{1}{2} \cdot 4 & - \frac{1}{2} \cdot 0 & - \frac{1}{2} \cdot 10 \end{array}]$

Multiply $- 1$ by $4$ .

$x = [\begin{array}{c} 1 & - 4 & - \frac{1}{2} \cdot 2 \\ - \frac{1}{2} \cdot 4 & - \frac{1}{2} \cdot 0 & - \frac{1}{2} \cdot 10 \end{array}]$

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{1}{2}$ into the numerator.

$x = [\begin{array}{c} 1 & - 4 & \frac{- 1}{2} \cdot 2 \\ - \frac{1}{2} \cdot 4 & - \frac{1}{2} \cdot 0 & - \frac{1}{2} \cdot 10 \end{array}]$

Cancel the common factor.

$x = [\begin{array}{c} 1 & - 4 & \frac{- 1}{2} \cdot 2 \\ - \frac{1}{2} \cdot 4 & - \frac{1}{2} \cdot 0 & - \frac{1}{2} \cdot 10 \end{array}]$

Rewrite the expression.

$x = [\begin{array}{c} 1 & - 4 & - 1 \\ - \frac{1}{2} \cdot 4 & - \frac{1}{2} \cdot 0 & - \frac{1}{2} \cdot 10 \end{array}]$

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{1}{2}$ into the numerator.

$x = [\begin{array}{c} 1 & - 4 & - 1 \\ \frac{- 1}{2} \cdot 4 & - \frac{1}{2} \cdot 0 & - \frac{1}{2} \cdot 10 \end{array}]$

Factor $2$ out of $4$ .

$x = [\begin{array}{c} 1 & - 4 & - 1 \\ \frac{- 1}{2} \cdot (2 (2)) & - \frac{1}{2} \cdot 0 & - \frac{1}{2} \cdot 10 \end{array}]$

Cancel the common factor.

$x = [\begin{array}{c} 1 & - 4 & - 1 \\ \frac{- 1}{2} \cdot (2 \cdot 2) & - \frac{1}{2} \cdot 0 & - \frac{1}{2} \cdot 10 \end{array}]$

Rewrite the expression.

$x = [\begin{array}{c} 1 & - 4 & - 1 \\ - 1 \cdot 2 & - \frac{1}{2} \cdot 0 & - \frac{1}{2} \cdot 10 \end{array}]$

Multiply $- 1$ by $2$ .

$x = [\begin{array}{c} 1 & - 4 & - 1 \\ - 2 & - \frac{1}{2} \cdot 0 & - \frac{1}{2} \cdot 10 \end{array}]$

Multiply $- \frac{1}{2} \cdot 0$ .

Multiply $0$ by $- 1$ .

$x = [\begin{array}{c} 1 & - 4 & - 1 \\ - 2 & 0 (\frac{1}{2}) & - \frac{1}{2} \cdot 10 \end{array}]$

Multiply $0$ by $\frac{1}{2}$ .

$x = [\begin{array}{c} 1 & - 4 & - 1 \\ - 2 & 0 & - \frac{1}{2} \cdot 10 \end{array}]$

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{1}{2}$ into the numerator.

$x = [\begin{array}{c} 1 & - 4 & - 1 \\ - 2 & 0 & \frac{- 1}{2} \cdot 10 \end{array}]$

Factor $2$ out of $10$ .

$x = [\begin{array}{c} 1 & - 4 & - 1 \\ - 2 & 0 & \frac{- 1}{2} \cdot (2 (5)) \end{array}]$

Cancel the common factor.

$x = [\begin{array}{c} 1 & - 4 & - 1 \\ - 2 & 0 & \frac{- 1}{2} \cdot (2 \cdot 5) \end{array}]$

Rewrite the expression.

$x = [\begin{array}{c} 1 & - 4 & - 1 \\ - 2 & 0 & - 1 \cdot 5 \end{array}]$

Multiply $- 1$ by $5$ .

$x = [\begin{array}{c} 1 & - 4 & - 1 \\ - 2 & 0 & - 5 \end{array}]$

Solve the Matrix Equation [[4,-3,2],[0,5,-8]]-2x=[[2,5,4],[4,5,2]]

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