Question

If \( A \) and \( B \) are square matrices of order \( m \) such that \( A^2 - B^2 = (A - B)(A + B) \), then which of the following is always correct?

• A. \( A = B \)
• B. \( AB = BA \)
• C. \( A = 0 \) or \( B = 0 \)
• D. \( A = I \) or \( B = I \)

Hint:

The difference of squares formula \( a^2 - b^2 = (a - b)(a + b) \) holds for matrices just as it does for numbers, but ensure that the matrix operations are valid.

Answer 1

The Correct Option is A

To solve the problem involving square matrices \( A \) and \( B \) of order \( m \) with the condition \( A^2 - B^2 = (A - B)(A + B) \), let's explore each step.

Recall the algebraic identity for the difference of squares:

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

This is true generally for matrices under the assumption that multiplication is commutative, but matrices don't generally commute. However, since the equation is given as a fact, it signals that either the matrices are commutative or some specific condition holds.

The structure suggests possible simplifications:

1. One simplification could be commutativity: if \( A \) and \( B \) commute, i.e., \( AB = BA \), the equation holds as is.

2. The trivial solution: if \( A = B \), then both sides evaluate to \( 0 \) as \( A - B = 0 \).

In this setup, the condition \( A = B \) always holds true under the given identity, irrespective of commutativity assumptions.

Thus, from the options provided, the correct answer is:

\( A = B \).