Class CBSE Class 12 Mathematics Matrices and Determinants Q #1488
KNOWLEDGE BASED
REMEMBER
1 Marks 2026 AISSCE(Board Exam) MCQ SINGLE
For any square matrix A with real entries, if $A + A'$ is a symmetric matrix then :
(A) (A - A') cannot be a skew symmetric matrix
(B) (A - A') is a skew symmetric matrix
(C) A is always a symmetric matrix
(D) A is always a skew symmetric matrix
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Correct Answer: B

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Detailed Solution

Step 1: Understanding Symmetric and Skew-Symmetric Matrices

A matrix $M$ is symmetric if $M' = M$, and skew-symmetric if $M' = -M$.

Step 2: Given Condition

We are given that $A + A'$ is a symmetric matrix. This means $(A + A')' = A + A'$.

Step 3: Analyzing (A - A')

Let's consider the transpose of $(A - A')$: $$(A - A')' = A' - (A')' = A' - A = -(A - A')$$ This shows that $(A - A')$ is a skew-symmetric matrix.

Step 4: Evaluating the Options

(A) (A - A') cannot be a skew symmetric matrix - This is incorrect, as we showed (A - A') is skew-symmetric. (B) (A - A') is a skew symmetric matrix - This is correct. (C) A is always a symmetric matrix - This is not necessarily true. (D) A is always a skew symmetric matrix - This is not necessarily true.

Final Answer: Option B<\/span>

AI Suggestion: Option B

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Pedagogical Audit
Bloom's Analysis: This is an REMEMBER question because it requires recalling the definitions of symmetric and skew-symmetric matrices and applying the properties of transpose.
Knowledge Dimension: CONCEPTUAL
Justification: The question tests the understanding of the concepts of symmetric and skew-symmetric matrices and their properties related to transpose.
Syllabus Audit: In the context of CBSE Class 12, this is classified as KNOWLEDGE. It directly relates to the definitions and properties of matrices covered in the textbook.

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