Question

If $A$ and $B$ are square matrices of same order, then which of the following statements is/are always true?
(i) $(A+B)(A-B)=A^2-B^2$
(ii) $AB=BA$
(iii) $(A+B)^2=A^2+AB+BA+B^2$
(iv) $AB=0 \Rightarrow A=0$ or $B=0$

• A. Only (i) and (iii)
• B. Only (ii) and (iii)
• C. Only (iii)
• D. Only (ii) and (iv)

Hint:

Matrix multiplication is generally \textbf{not commutative}. Always remember that \(AB \ne BA\) in most cases.

Answer 1

The Correct Option is C

To evaluate which statements are always true when dealing with square matrices \( A \) and \( B \) of the same order, let's examine each statement one-by-one:

(i) \((A+B)(A-B)=A^2-B^2\)

In general, for matrices, the identity \((A+B)(A-B) = A^2 - B^2\) does not hold because matrix multiplication is not commutative. Specifically, \((A+B)(A-B)\) expands to:

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

As seen, this expression simplifies to \(A^2 - B^2\) only if \(AB = BA\), which is not always true for arbitrary matrices. Therefore, statement (i) is not always true.

(ii) \(AB = BA\)

In matrix algebra, the commutative property \(AB = BA\) does not generally hold for arbitrary matrices \(A\) and \(B\). This property holds for specific cases, such as when matrices are diagonal or mutually compatible, but not generally. Therefore, statement (ii) is not always true.

(iii) \((A+B)^2 = A^2 + AB + BA + B^2\)

This is true for matrices since:

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

Here, matrix products \(AB\) and \(BA\) naturally derive from the expansion, meaning statement (iii) is always true for any square matrices \(A\) and \(B\).

(iv) \(AB=0 \Rightarrow A=0\) or \(B=0\)

This property of matrices states that if the product of two matrices \( AB \) is a zero matrix, it does not imply that either \( A \) or \( B \) is a zero matrix. There exist non-zero matrices \( A \) and \( B \) such that \( AB = 0 \). Hence, statement (iv) is not always true.

Based on this analysis, only statement (iii) is always true. Thus, the correct answer is:

Only (iii)