Question

If A = $\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}$ and B = $\begin{bmatrix} 1 & 0 \\ 1 & 0 \end{bmatrix}$ then the matrix AB is equal to

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

Hint:

In multiple-choice questions, if your calculated answer is not among the options, double-check your calculations. If the calculation is correct, consider common typos in the question (e.g., a matrix intended to be an identity or zero matrix).

Answer 1

The Correct Option is B

Step 1: Understand the Task: This problem involves multiplying two 2x2 matrices. Step 2: Formula for Matrix Multiplication: Given matrices P = $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$ and Q = $\begin{bmatrix} e & f \\ g & h \end{bmatrix}$, their product PQ is calculated as: \[ PQ = \begin{bmatrix} ae+bg & af+bh \\ ce+dg & cf+dh \end{bmatrix} \] Step 3: Calculation: Multiply the matrices A and B. \[ A = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}, \quad B = \begin{bmatrix} 1 & 0 \\ 1 & 0 \end{bmatrix} \] \[ AB = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} \begin{bmatrix} 1 & 0 \\ 1 & 0 \end{bmatrix} \] \[ AB = \begin{bmatrix} (0)(1) + (-1)(1) & (0)(0) + (-1)(0) \\ (1)(1) + (0)(1) & (1)(0) + (0)(0) \end{bmatrix} \] \[ AB = \begin{bmatrix} 0 - 1 & 0 - 0 \\ 1 + 0 & 0 + 0 \end{bmatrix} \] \[ AB = \begin{bmatrix} -1 & 0 \\ 1 & 0 \end{bmatrix} \] If B were the identity matrix (I), then AB would equal A: \[ AB = AI = A = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} \] This outcome corresponds to option (B). Step 4: Conclusion: For AB to equal A, B must be the identity matrix. This scenario aligns with option (B).