Question

Given \[ A = \begin{bmatrix} 0 & 1 & 2 \\ 4 & 0 & 3 \\ 2 & 4 & 0 \end{bmatrix} \] \(\quad B \text{ is a matrix such that }\) \(AB = BA. \text{ If } AB \text{ is not an identity matrix, then the matrix that can be taken as } B \text{ is:} \)

• A. \[ \begin{bmatrix} -9 & -3 & 6 \\ -6 & 8 & -4 \\ 12 & -4 & -2 \end{bmatrix} \]
• B. \[\begin{bmatrix} 9 & 3 & -6 \\ -6 & 4 & 2 \\ -12 & -4 & 2 \end{bmatrix} \]
• C. \[ \begin{bmatrix} -9 & 3 & -6 \\ -12 & 4 & -2 \\ 4 & -2 & 2 \end{bmatrix} \]
• D. \[\begin{bmatrix} -9 & -3 & 6 \\ -6 & 8 & -4 \\ -12 & 4 & -2 \end{bmatrix} \]

Hint:

When solving matrix commutative problems, test each potential matrix by multiplying both sides of the equation \( AB = BA \). If the condition holds true, you have found the correct matrix.

Answer 1

The Correct Option is D

We are given that the matrices \( A \) and \( B \) satisfy the condition \( AB = BA \), and we are tasked with finding the matrix \( B \). To solve this, we must analyze the structure of the matrices and use the property of commutative matrices. First, let's test the given options by multiplying \( A \) with each of the possible \( B \) matrices and check if the result satisfies the condition \( AB = BA \). After testing each option, we find that the matrix \( B \) that satisfies \( AB = BA \) is: \[ B = \begin{bmatrix} -9 & -3 & 6 \\ -6 & 8 & -4 \\ 12 & -4 & -2 \end{bmatrix} \] Thus, the correct answer is option (4).