Question

Let A and B be two 2 × 2 matrices. Consider the statements

(i) AB = O Ž A = O or B = O

(ii) AB = I₂Ž A = B^–1

(iii) (A + B)² = A² + 2AB + B²

Then

• A. (i) is false, (ii) and (iii) are true
• B. (i) and (iii) are false, (ii) is true
• C. (i) and (ii) are false, (iii) is true
• D. (ii) and (iii) are false, (i) is true

Answer 1

Answer: (B)

(i) and (iii) are false, (ii) is true

Answer 2

(i) is false,

If A = $\begin{bmatrix} 0 & 1 \\ 0 & - 1 \end{bmatrix}$ and B = $\begin{bmatrix} 1 & 1 \\ 0 & 0 \end{bmatrix}$, then AB = $\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$ = O

Thus, AB = 0 Ž A = O or B = O

(iii) is false since matrix multiplication is not commutative.

(ii) is true as product AB is an identity matrix, if B is inverse of the matrix A.