Question

If \( 2A + 3B = \begin{bmatrix} 2 & -1 & 4 \\ 3 & 2 & 5 \end{bmatrix} \) and \( A + 2B = \begin{bmatrix} 5 & 0 & 3 \\ 1 & 6 & 2 \end{bmatrix} \), then \( B = \)

• A. \( \begin{bmatrix} -8 & -1 & -2 \\ 1 & -10 & 1 \end{bmatrix} \)
• B. \( \begin{bmatrix} 8 & 1 & -2 \\ 1 & 10 & -1 \end{bmatrix} \)
• C. \( \begin{bmatrix} 8 & 1 & 2 \\ -1 & 10 & -1 \end{bmatrix} \)
• D. \( \begin{bmatrix} 8 & -1 & 2 -1 & 10 & -1 \end{bmatrix} \)

Hint:

In matrix equations, to isolate a matrix variable, manipulate the equations by scaling or adding/subtracting them as neede(D)

Answer 1

The Correct Option is C

We are given the system of matrix equations: \[ 2A + 3B = \begin{bmatrix} 2 & -1 & 4 \\ 3 & 2 & 5 \end{bmatrix} \] and \[ A + 2B = \begin{bmatrix} 5 & 0 & 3 \\ 1 & 6 & 2 \end{bmatrix} \] Step 1: Solve for \( A \) and \( B \) We have two equations and two unknowns, so we can solve for \( A \) and \( B \). First, let's multiply the second equation by 2: \[ 2A + 4B = \begin{bmatrix} 10 & 0 & 6
2 & 12 & 4 \end{bmatrix} \] Now, subtract the first equation from this new equation: \[ (2A + 4B) - (2A + 3B) = \begin{bmatrix} 10 & 0 & 6 \\ 2 & 12 & 4 \end{bmatrix} - \begin{bmatrix} 2 & -1 & 4 \\ 3 & 2 & 5 \end{bmatrix} \] Simplifying both sides: \[ B = \begin{bmatrix} 8 & 1 & 2 \\ -1 & 10 & -1 \end{bmatrix} \] Thus, the value of \( B \) is \( \begin{bmatrix} 8 & 1 & 2 \\ -1 & 10 & -1 \end{bmatrix} \).