Question

If \( A \) and \( B \) are both \( 3 \times 3 \) matrices, then which of the following statements are true? \[ \begin{cases}

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

Hint:

Remember matrix multiplication is not commutative: \( AB \neq BA \) in general.

Answer 1

The Correct Option is B

Step 1: Analyze (i)
If \( AB = 0 \), it does not necessarily imply \( A = 0 \) or \( B = 0 \). Non-zero matrices can multiply to zero. Hence, (i) is false. Step 2: Analyze (ii)
If \( AB = I_3 \), then \( B \) is the right inverse of \( A \). For square matrices, right inverse equals inverse, so \( A^{-1} = B \). Hence, (ii) is true. Step 3: Analyze (iii)
For matrices, \( (A-B)^2 = A^2 - AB - BA + B^2 \), not \( A^2 - 2AB + B^2 \) because matrix multiplication is generally not commutative. Hence, (iii) is false.