Class CBSE Class 12 Mathematics Matrices and Determinants Q #760
KNOWLEDGE BASED
APPLY
1 Marks 2025 MCQ SINGLE
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 \(B=I\)
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Correct Answer: B
Explanation


The identity $A^2 - B^2 = (A - B)(A + B)$ holds for square matrices $A$ and $B$
only if they commute, i.e., $AB = BA$.
This is because matrix multiplication is not generally commutative.

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Step-by-Step Solution

  1. Given: \(A^{2}-B^{2}=(A-B)(A+B)\)
  2. Expanding the right side: \((A-B)(A+B) = A^2 + AB - BA - B^2\)
  3. So, \(A^{2}-B^{2} = A^2 + AB - BA - B^2\)
  4. Subtracting \(A^2\) and adding \(B^2\) to both sides: \(0 = AB - BA\)
  5. Therefore, \(AB = BA\)

Correct Answer: \(AB=BA\)

AI Suggestion: Option B

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Pedagogical Audit
Bloom's Analysis: This is an APPLY question because the student needs to apply the properties of matrix multiplication to determine the correct relationship between matrices A and B.
Knowledge Dimension: CONCEPTUAL
Justification: The question requires understanding the concepts of matrix algebra, specifically matrix multiplication and the conditions under which the given equation holds true. It's not just about recalling facts but applying the rules of matrix operations.
Syllabus Audit: In the context of CBSE Class 12, this is classified as KNOWLEDGE. The question directly tests the student's understanding of matrix algebra properties as covered in the textbook.

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