Class CBSE Class 12 Mathematics Matrices and Determinants Q #1675
COMPETENCY BASED
APPLY
1 Marks 2026 AISSCE(Board Exam) MCQ SINGLE
If a square matrix A is such that $A^{2}=A$ and $(I-A)^{3}=xA+I$ then value of x must be:
(A) 7
(B) 5
(C) -7
(D) -1
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Correct Answer: D

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Detailed Solution

Step 1: Expand the expression

We are given the equation $$(I-A)^3 = xA + I$$. Using the binomial expansion for matrices (since $I$ and $A$ commute, i.e., $IA = AI$), we have: $$(I-A)^3 = I^3 - 3I^2A + 3IA^2 - A^3$$ Since $I^n = I$ and $IA = AI = A$, this simplifies to: $$I - 3A + 3A^2 - A^3 = xA + I$$

Step 2: Apply the idempotent property

We are given that $A^2 = A$. We can use this to simplify higher powers of $A$: $$A^3 = A^2 \cdot A = A \cdot A = A^2 = A$$ Substituting $A^2 = A$ and $A^3 = A$ into the expanded equation: $$I - 3A + 3(A) - (A) = xA + I$$

Step 3: Solve for x

Simplify the left side of the equation: $$I - 3A + 3A - A = xA + I$$ $$I - A = xA + I$$ Subtracting $I$ from both sides: $$-A = xA$$ Comparing the coefficients of $A$, we get: $$x = -1$$

Final Answer: -1

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Pedagogical Audit
Bloom's Analysis: This is an APPLY question because the student must apply the properties of idempotent matrices and binomial expansion in a matrix context to solve for an unknown variable.
Knowledge Dimension: PROCEDURAL
Justification: The question requires a step-by-step algebraic manipulation of matrix equations, testing the student's ability to execute a sequence of logical operations.
Syllabus Audit: In the context of CBSE Class 12, this is classified as COMPETENCY. It tests the conceptual understanding of matrix algebra beyond simple arithmetic, aligning with NEP 2020's focus on higher-order thinking skills.
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