Question

If \( A \) and \( B \) are both \( 3 \times 3 \) matrices, then which of the following statements are true?
(i) \( AB = 0 \Rightarrow A = 0 \) or \( B = 0 \)
(ii) \( AB = I_3 \Rightarrow A^{-1} = B \)
(iii) \( (A - B)^2 = A^2 - 2AB + B^2 \)

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

Hint:

To avoid common pitfalls in matrix algebra, always treat the order of multiplication as fixed. Since \( AB \) is usually not equal to \( BA \), standard algebraic identities like \( (a-b)^2 \) must be expanded manually as \( A^2 - AB - BA + B^2 \).

Answer 1

The Correct Option is B

To solve the given problem, we need to analyze each of the statements related to \(3 \times 3\) matrices \(A\) and \(B\). Let's evaluate them one by one:

Statement (i): \( AB = 0 \Rightarrow A = 0 \) or \( B = 0 \)

In general, the statement \( AB = 0 \) does not imply that \( A = 0 \) or \( B = 0 \). This can be true for matrices that are not zero matrices. Consider the following example of \(2 \times 2\) matrices:

Let \( A = \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix} \) and \( B = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix} \). Then,

1. \(AB = \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix} \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix} = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix} = 0\)
    

Here, neither \(A\) nor \(B\) is zero, but their product is the zero matrix. Hence, statement (i) is false.

Statement (ii): \( AB = I_3 \Rightarrow A^{-1} = B \)

If the product \(AB\) is the identity matrix \(I_3\), then by definition, \(B\) is the inverse of \(A\). Therefore, \(A^{-1} = B\) holds true because \(AB = I_3\) implies that \(A\) and \(B\) are invertible matrices and are inverses of each other. Hence, statement (ii) is true.

Statement (iii): \( (A - B)^2 = A^2 - 2AB + B^2 \)

Let's perform the algebraic expansion of \((A - B)^2\):

1. \((A - B)^2 = (A - B)(A - B) = A^2 - AB - BA + B^2\)
    

The correct expansion is \(A^2 - AB - BA + B^2\), which differs from \(A^2 - 2AB + B^2\). Note that in general, matrix multiplication is not commutative, i.e., \(AB \neq BA\), so we cannot combine \(-AB\) and \(-BA\) into \(-2AB\). Thus, statement (iii) is false.

Based on the above analysis, only statement (ii) is true, whereas (i) and (iii) are false.

Conclusion: The correct answer is: (ii) is true (i), (iii) are false.