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

Concept: Matrix algebra features several distinct properties: the product of two non-zero matrices can be zero (zero divisors), and matrix multiplication is generally non-commutative (\( AB \neq BA \)). For square matrices, if their product is the identity matrix (\( I \)), then one is the inverse of the other.

Step 1: Checking statement (i) regarding zero divisors.
In matrix algebra, \( AB = 0 \) does not necessarily mean \( A = 0 \) or \( B = 0 \). There exist non-zero matrices whose product is zero. For example: \[ A = \begin{pmatrix} 0 & 1 0 & 0 \end{pmatrix}, B = \begin{pmatrix} 1 & 0 0 & 0 \end{pmatrix} \Rightarrow AB = \begin{pmatrix} 0 & 0 0 & 0 \end{pmatrix} \] Since \( A, B \neq 0 \) but \( AB = 0 \), statement (i) is false.

Step 2: Checking statement (ii) regarding the matrix inverse.
By the definition of an invertible matrix, if \( A \) and \( B \) are square matrices of the same order such that \( AB = I \), then \( A \) is non-singular and its inverse \( A^{-1} \) is equal to \( B \). Thus, statement (ii) is true.

Step 3: Checking statement (iii) regarding the expansion of \((A-B)^2\).
We expand the square by multiplying the matrices: \[ (A - B)^2 = (A - B)(A - B) = A^2 - AB - BA + B^2 \] This expression only simplifies to \( A^2 - 2AB + B^2 \) if \( AB = BA \). Because matrix multiplication is not generally commutative, statement (iii) is false.