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 address the problem with square matrices \( A \) and \( B \) of order \( m \) where \( A^2 - B^2 = (A - B)(A + B) \), we will examine each stage.

The standard algebraic identity for the difference of squares is:

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

This identity is generally true for matrices if multiplication is commutative. However, matrix multiplication is not always commutative. The fact that the equation is provided as true suggests either the matrices commute or a specific condition is met.

The problem's structure indicates potential simplifications:

1. Commutativity is one possibility: If \( A \) and \( B \) commute, meaning \( AB = BA \), the given equation holds true.

2. The trivial solution is another: If \( A = B \), then \( A - B = 0 \), resulting in \( 0 \) on both sides of the equation.

Within this context, the condition \( A = B \) is universally true under the given identity, regardless of whether the matrices commute.

Therefore, from the available choices, the accurate conclusion is:

\( A = B \).