Solve the Matrix Equation [[6,8],[-11,15]]-[[c+2,3],[-5,d-4]]=[[22,5],[-6,17]]

$[\begin{array}{c} 6 & 8 \\ - 11 & 15 \end{array}] - [\begin{array}{c} c + 2 & 3 \\ - 5 & d - 4 \end{array}] = [\begin{array}{c} 22 & 5 \\ - 6 & 17 \end{array}]$

Step 1

Subtract the matrix from both sides of the equation.

$[\begin{array}{c} 6 & 8 \\ - 11 & 15 \end{array}] - X - [\begin{array}{c} 6 & 8 \\ - 11 & 15 \end{array}] = [\begin{array}{c} 22 & 5 \\ - 6 & 17 \end{array}] - [\begin{array}{c} 6 & 8 \\ - 11 & 15 \end{array}]$

Step 2

Simplify both sides of the equation.

Simplify the left side of the equation.

Subtract the corresponding elements.

$[\begin{array}{c} 6 - 6 & 8 - 8 \\ - 11 - (- 11) & 15 - 15 \end{array}] - X = [\begin{array}{c} 22 & 5 \\ - 6 & 17 \end{array}] - [\begin{array}{c} 6 & 8 \\ - 11 & 15 \end{array}]$

Simplify each element of the matrix $[\begin{array}{c} 6 - 6 & 8 - 8 \\ - 11 - (- 11) & 15 - 15 \end{array}]$ .

Simplify $6 - 6$ .

$[\begin{array}{c} 0 & 8 - 8 \\ - 11 - (- 11) & 15 - 15 \end{array}] - X = [\begin{array}{c} 22 & 5 \\ - 6 & 17 \end{array}] - [\begin{array}{c} 6 & 8 \\ - 11 & 15 \end{array}]$

Simplify $8 - 8$ .

$[\begin{array}{c} 0 & 0 \\ - 11 - (- 11) & 15 - 15 \end{array}] - X = [\begin{array}{c} 22 & 5 \\ - 6 & 17 \end{array}] - [\begin{array}{c} 6 & 8 \\ - 11 & 15 \end{array}]$

Simplify $- 11 - (- 11)$ .

$[\begin{array}{c} 0 & 0 \\ 0 & 15 - 15 \end{array}] - X = [\begin{array}{c} 22 & 5 \\ - 6 & 17 \end{array}] - [\begin{array}{c} 6 & 8 \\ - 11 & 15 \end{array}]$

Simplify $15 - 15$ .

$[\begin{array}{c} 0 & 0 \\ 0 & 0 \end{array}] - X = [\begin{array}{c} 22 & 5 \\ - 6 & 17 \end{array}] - [\begin{array}{c} 6 & 8 \\ - 11 & 15 \end{array}]$

Simplify the right side of the equation.

Subtract the corresponding elements.

$[\begin{array}{c} 0 & 0 \\ 0 & 0 \end{array}] - X = [\begin{array}{c} 22 - 6 & 5 - 8 \\ - 6 - (- 11) & 17 - 15 \end{array}]$

Simplify each element of the matrix $[\begin{array}{c} 22 - 6 & 5 - 8 \\ - 6 - (- 11) & 17 - 15 \end{array}]$ .

Simplify $22 - 6$ .

$[\begin{array}{c} 0 & 0 \\ 0 & 0 \end{array}] - X = [\begin{array}{c} 16 & 5 - 8 \\ - 6 - (- 11) & 17 - 15 \end{array}]$

Simplify $5 - 8$ .

$[\begin{array}{c} 0 & 0 \\ 0 & 0 \end{array}] - X = [\begin{array}{c} 16 & - 3 \\ - 6 - (- 11) & 17 - 15 \end{array}]$

Simplify $- 6 - (- 11)$ .

$[\begin{array}{c} 0 & 0 \\ 0 & 0 \end{array}] - X = [\begin{array}{c} 16 & - 3 \\ 5 & 17 - 15 \end{array}]$

Simplify $17 - 15$ .

$[\begin{array}{c} 0 & 0 \\ 0 & 0 \end{array}] - X = [\begin{array}{c} 16 & - 3 \\ 5 & 2 \end{array}]$

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

$- X = [\begin{array}{c} 16 & - 3 \\ 5 & 2 \end{array}]$

Step 3

Multiply both sides of the equation by $- 1$ .

$- 1 \cdot (- X) = - 1 \cdot [\begin{array}{c} 16 & - 3 \\ 5 & 2 \end{array}]$

Step 4

Simplify both sides of the equation.

Multiply $- 1 (- X)$ .

Multiply $- 1$ by $- 1$ .

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

Multiply $X$ by $1$ .

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

Simplify the right side of the equation.

Multiply $- 1$ by each element of the matrix.

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

Simplify each element in the matrix.

Rearrange $- 1 \cdot 16$ .

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

Rearrange $- 1 \cdot - 3$ .

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

Rearrange $- 1 \cdot 5$ .

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

Rearrange $- 1 \cdot 2$ .

$X = [\begin{array}{c} - 16 & 3 \\ - 5 & - 2 \end{array}]$

Solve the Matrix Equation [[6,8],[-11,15]]-[[c+2,3],[-5,d-4]]=[[22,5],[-6,17]]

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