Class CBSE Class 12 Mathematics Matrices and Determinants Q #1689
COMPETENCY BASED
APPLY
1 Marks 2026 AISSCE(Board Exam) MCQ SINGLE
If A and B are skew-symmetric matrices of same order, then $AB^{\prime}+BA^{\prime}$ is a/an:
(A) symmetric matrix
(B) skew-symmetric matrix
(C) null matrix
(D) identity matrix
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Correct Answer: A

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Detailed Solution

Step 1: Identify properties of skew-symmetric matrices

Given that A and B are skew-symmetric matrices, we have the properties: $$A' = -A$$ $$B' = -B$$

Step 2: Define the matrix expression

Let $X = AB' + BA'$. We need to check if $X$ is symmetric or skew-symmetric by finding its transpose $X'$.

Step 3: Apply transpose properties

Using the property $(P+Q)' = P' + Q'$ and $(PQ)' = Q'P'$: $$X' = (AB' + BA')' = (AB')' + (BA')'$$ $$X' = (B')'A' + (A')'B'$$ Since $(M')' = M$, we get: $$X' = BA' + AB'$$

Step 4: Substitute skew-symmetric properties

Substitute $A' = -A$ and $B' = -B$ into the expression: $$X' = B(-A) + A(-B)$$ $$X' = -BA - AB = -(AB + BA)$$ Wait, let us re-evaluate the original expression $AB' + BA'$. Substitute $A' = -A$ and $B' = -B$: $$X = A(-B) + B(-A) = -AB - BA = -(AB + BA)$$ Now find $X'$: $$X' = (-(AB + BA))' = -(B'A' + A'B')$$ $$X' = -((-B)(-A) + (-A)(-B)) = -(BA + AB) = X$$ Since $X' = X$, the matrix is symmetric.

Final Answer: symmetric matrix

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Pedagogical Audit
Bloom's Analysis: This is an APPLY question because the student must utilize the definitions and properties of transpose and skew-symmetric matrices to derive the nature of a composite matrix expression.
Knowledge Dimension: PROCEDURAL
Justification: The question requires a step-by-step algebraic manipulation of matrix properties rather than simple recall of facts.
Syllabus Audit: In the context of CBSE Class 12, this is classified as COMPETENCY. It tests the student's ability to apply theoretical properties of matrices to solve abstract problems, which is a core requirement of the Matrices chapter in the NCERT curriculum.

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