QuestionSolve the matrix equation $2 X+A=B$ for $X$ if $A=\left[\begin{array}{rr}3 & -8 \\ 0 & 9 \\ -5 & 9\end{array}\right]$ and $B=\left[\begin{array}{rr}-1 & -8 \\ 8 & 4 \\ -3 & 3\end{array}\right]$ . $X=$ $\square$ (Simplify your answer.)

Studdy Solution

STEP 1

1. We are given a matrix equation $2X + A = B$ .
2. Matrices $A$ and $B$ are provided.
3. We need to solve for the matrix $X$ .

STEP 2

1. Isolate the matrix $X$ .
2. Substitute the given matrices $A$ and $B$ .
3. Perform matrix subtraction.
4. Divide the resulting matrix by 2 to solve for $X$ .

STEP 3

To isolate $X$ , subtract matrix $A$ from both sides of the equation:
$2X + A = B$ $2X = B - A$

STEP 4

Substitute the given matrices $A$ and $B$ into the equation:
$A = \begin{bmatrix} 3 & -8 \\ 0 & 9 \\ -5 & 9 \end{bmatrix}, \quad B = \begin{bmatrix} -1 & -8 \\ 8 & 4 \\ -3 & 3 \end{bmatrix}$
$2X = \begin{bmatrix} -1 & -8 \\ 8 & 4 \\ -3 & 3 \end{bmatrix} - \begin{bmatrix} 3 & -8 \\ 0 & 9 \\ -5 & 9 \end{bmatrix}$

STEP 5

Perform matrix subtraction:
$2X = \begin{bmatrix} -1 - 3 & -8 - (-8) \\ 8 - 0 & 4 - 9 \\ -3 - (-5) & 3 - 9 \end{bmatrix}$
$2X = \begin{bmatrix} -4 & 0 \\ 8 & -5 \\ 2 & -6 \end{bmatrix}$

STEP 6

Divide each element of the resulting matrix by 2 to solve for $X$ :
$X = \frac{1}{2} \begin{bmatrix} -4 & 0 \\ 8 & -5 \\ 2 & -6 \end{bmatrix}$
$X = \begin{bmatrix} -2 & 0 \\ 4 & -\frac{5}{2} \\ 1 & -3 \end{bmatrix}$
The solution for $X$ is:
$X = \begin{bmatrix} -2 & 0 \\ 4 & -\frac{5}{2} \\ 1 & -3 \end{bmatrix}$

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