Class CBSE Class 12 Mathematics Matrices and Determinants Q #1690
COMPETENCY BASED
APPLY
1 Marks 2026 AISSCE(Board Exam) MCQ SINGLE
If A and B are square matrices of same order, then which of the following statements is/are always true? (i) $(A+B)(A-B)=A^{2}-B^{2}$ (ii) $AB=BA$ (iii) $(A+B)^{2}=A^{2}+AB+BA+B^{2}$ (iv) $AB=0 \Rightarrow A=0$ or $B=0$
(A) Only (i) and (iii)
(B) Only (ii) and (iii)
(C) Only (iii)
(D) Only (iii) and (iv)
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Detailed Solution

Step 1: Analyze Statement (i)

The expression $(A+B)(A-B)$ expands to $A^2 - AB + BA - B^2$. Since matrix multiplication is generally not commutative ($AB \neq BA$), we cannot simplify $-AB + BA$ to $0$. Thus, $(A+B)(A-B) = A^2 - B^2$ is only true if $AB = BA$. It is not always true.

Step 2: Analyze Statement (ii)

Matrix multiplication is not commutative in general. Therefore, $AB = BA$ is not always true for any two square matrices $A$ and $B$.

Step 3: Analyze Statement (iii)

Expanding $(A+B)^2$ means $(A+B)(A+B)$. Using the distributive property: $$ (A+B)(A+B) = A(A+B) + B(A+B) = A^2 + AB + BA + B^2 $$ This is a fundamental property of matrix algebra and is always true.

Step 4: Analyze Statement (iv)

The zero-product property does not hold for matrices. For example, if $A = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$ and $B = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}$, then $AB = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}$, but neither $A$ nor $B$ is the zero matrix. Thus, this is false.

Final Answer: Only (iii)

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Pedagogical Audit
Bloom's Analysis: This is an APPLY question because the student must apply the properties of matrix algebra to verify the validity of specific algebraic identities in a non-commutative context.
Knowledge Dimension: CONCEPTUAL
Justification: The question tests the understanding of fundamental matrix properties, specifically non-commutativity and the zero-divisor property.
Syllabus Audit: In the context of CBSE Class 12, this is classified as COMPETENCY. It moves beyond rote memorization of formulas to test the conceptual boundaries of matrix operations.

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