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?
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?
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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.
Updated On: Mar 8, 2025
- \( A = B \)
- \( AB = BA \)
- \( A = 0 \) or \( B = 0 \)
- \( A = I \) or \( B = I \)
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The Correct Option is A
Solution and Explanation
Using the difference of squares formula, we know:
\[
A^2 - B^2 = (A - B)(A + B)
\]
For this to hold, it must be true that \( A = B \) because otherwise, the matrices \( (A - B) \) and \( (A + B) \) would not satisfy the equation for all cases. Thus, the correct answer is \( A = B \).
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