Question

If \[ A=\begin{bmatrix} 3 & 4\\ 5 & 6 \end{bmatrix} \] and \[ B=\begin{bmatrix} x & 0\\ 0 & y \end{bmatrix}, \quad x,y\in \mathbb{N}, \] then:

• A. There is exactly one such matrix \(B\) such that \(AB=I\)
• B. There is no matrix \(B\) such that \(AB=BA\)
• C. There exist only a finite number of matrices \(B\) such that \(AB=BA\)
• D. There exist infinite number of matrices \(B\) such that \(AB=BA\)

Hint:

For matrix commutativity questions, always compute both \(AB\) and \(BA\), then compare corresponding entries.

Answer 1

The Correct Option is D

Step 1: Write the given matrices.
We are given \[ A=\begin{bmatrix} 3 & 4\\ 5 & 6 \end{bmatrix} \] and \[ B=\begin{bmatrix} x & 0\\ 0 & y \end{bmatrix} \] where \[ x,y\in \mathbb{N} \]

Step 2: Find \(AB\).
\[ AB= \begin{bmatrix} 3 & 4\\ 5 & 6 \end{bmatrix} \begin{bmatrix} x & 0\\ 0 & y \end{bmatrix} \] Multiplying the matrices, \[ AB= \begin{bmatrix} 3x & 4y\\ 5x & 6y \end{bmatrix} \]

Step 3: Find \(BA\).
\[ BA= \begin{bmatrix} x & 0\\ 0 & y \end{bmatrix} \begin{bmatrix} 3 & 4\\ 5 & 6 \end{bmatrix} \] Multiplying the matrices, \[ BA= \begin{bmatrix} 3x & 4x\\ 5y & 6y \end{bmatrix} \]

Step 4: Apply the condition \(AB=BA\).
For \[ AB=BA, \] we must have \[ \begin{bmatrix} 3x & 4y\\ 5x & 6y \end{bmatrix} = \begin{bmatrix} 3x & 4x\\ 5y & 6y \end{bmatrix} \] Comparing corresponding elements, we get \[ 4y=4x \] and \[ 5x=5y \] Thus, \[ x=y \]

Step 5: Count the number of possible matrices.
Since \[ x,y\in \mathbb{N} \] and \[ x=y, \] we can take \[ x=y=1,2,3,4,\ldots \] So possible matrices are \[ B= \begin{bmatrix} 1 & 0\\ 0 & 1 \end{bmatrix}, \begin{bmatrix} 2 & 0\\ 0 & 2 \end{bmatrix}, \begin{bmatrix} 3 & 0\\ 0 & 3 \end{bmatrix}, \ldots \] Hence, there are infinitely many such matrices \(B\).

Step 6: Final conclusion.
Therefore, \[ \boxed{\text{There exist infinite number of matrices \(B\) such that \(AB=BA\)}} \]