Question

If \[ A = \begin{bmatrix} 8 & 0 \\ 4 & -2 \\ 3 & 6 \end{bmatrix}, B = \begin{bmatrix} 2 & -2 \\ 4 & 2 \\ -5 & 1 \end{bmatrix}, \] and \[ 2A + 3X = 5B, \text{then find the matrix} \, X. \]

Hint:

To solve for a matrix in an equation, isolate the matrix and perform matrix operations like addition, subtraction, and scalar multiplication.

Answer 1

We are given the matrix equation: \[ 2A + 3X = 5B. \]

Step 1: Isolate \( X \).
First, subtract \( 2A \) from both sides to isolate \( 3X \): \[ 3X = 5B - 2A. \] Now, multiply both sides by \( \frac{1}{3} \) to solve for \( X \): \[ X = \frac{1}{3}(5B - 2A). \]

Step 2: Calculate \( 5B - 2A \).
First, calculate \( 5B \) and \( 2A \): \[ 5B = 5 \begin{bmatrix} 2 & -2 \\ 4 & 2 \\ -5 & 1 \end{bmatrix} = \begin{bmatrix} 10 & -10 \\ 20 & 10 \\ -25 & 5 \end{bmatrix}, \] \[ 2A = 2 \begin{bmatrix} 8 & 0 \\ 4 & -2 \\ 3 & 6 \end{bmatrix} = \begin{bmatrix} 16 & 0 \\ 8 & -4 \\ 6 & 12 \end{bmatrix}. \] Now subtract \( 2A \) from \( 5B \): \[ 5B - 2A = \begin{bmatrix} 10 & -10 \\ 20 & 10 \\ -25 & 5 \end{bmatrix} - \begin{bmatrix} 16 & 0 \\ 8 & -4 \\ 6 & 12 \end{bmatrix} = \begin{bmatrix} -6 & -10 \\ 12 & 14 \\ -31 & -7 \end{bmatrix}. \]

Step 3: Find \( X \).
Now, divide each element of the matrix by 3: \[ X = \frac{1}{3} \begin{bmatrix} -6 & -10 \\ 12 & 14 \\ -31 & -7 \end{bmatrix} = \begin{bmatrix} -2 & -\frac{10}{3} \\ 4 & \frac{14}{3} \\ -\frac{31}{3} & -\frac{7}{3} \end{bmatrix}. \]
Conclusion:
The matrix \( X \) is: \[ X = \begin{bmatrix} -2 & -\frac{10}{3} \\ 4 & \frac{14}{3} \\ -\frac{31}{3} & -\frac{7}{3} \end{bmatrix}. \]