Class CBSE Class 12 Mathematics Matrices and Determinants Q #1683
COMPETENCY BASED
APPLY
1 Marks 2026 AISSCE(Board Exam) MCQ SINGLE
For a square matrix A, $(3A)^{-1}=$
(A) $3A^{-1}$
(B) $9A^{-1}$
(C) $\frac{1}{3}A^{-1}$
(D) $\frac{1}{9}A^{-1}$
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Correct Answer: C

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Detailed Solution

Step 1: Recall the property of scalar multiplication with inverse

For any non-singular square matrix A of order n and a non-zero scalar k, the property of the inverse of a product is given by (kA)-1 = k-1A-1.

Step 2: Apply the property to the given expression

In the given expression (3A)-1, the scalar k is 3. Applying the property:

$$ (3A)^{-1} = 3^{-1} A^{-1} $$

Step 3: Simplify the scalar

Since 3-1 = 1/3, we substitute this back into the expression:

$$ (3A)^{-1} = \frac{1}{3} A^{-1} $$

Final Answer: $\frac{1}{3}A^{-1}$

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Pedagogical Audit
Bloom's Analysis: This is an APPLY question because the student must identify and correctly apply the algebraic property of matrix inverses involving scalar multiplication.
Knowledge Dimension: PROCEDURAL
Justification: The question tests the student's ability to execute a specific mathematical procedure/rule regarding matrix operations.
Syllabus Audit: In the context of CBSE Class 12, this is classified as COMPETENCY. This question assesses conceptual clarity of the properties of determinants and inverses, which is a core competency in the Matrices and Determinants unit.

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