Question

If $A = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$ and $B = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}$, then $BA$ will be:

• A. $\begin{bmatrix} -1 & 0
  0 & 1 \end{bmatrix}$
• B. $\begin{bmatrix} 0 & -1
  1 & 0 \end{bmatrix}$
• C. $\begin{bmatrix} 0 & -1
  -1 & 0 \end{bmatrix}$
• D. $\begin{bmatrix} 0 & 1
  -1 & 0 \end{bmatrix}$

Hint:

Always check the order when multiplying matrices: $BA \neq AB$ in general.

Answer 1

The Correct Option is B

Step 1: Write the given matrices.
\[ A = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}, B = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} \]

Step 2: Compute $BA$. \\ \[ BA = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \] Performing multiplication: \[ BA = \begin{bmatrix} (1 \cdot 0 + 0 \cdot 1) & (1 \cdot 1 + 0 \cdot 0) \\ (0 \cdot 0 + (-1) \cdot 1) & (0 \cdot 1 + (-1) \cdot 0) \end{bmatrix} = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix} \]

Step 3: Compare with given options.
This matches option (B).

Step 4: Conclusion.
The correct answer is (B) $\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}$.