Class CBSE Class 12 Mathematics Matrices and Determinants Q #742
KNOWLEDGE BASED
APPLY
1 Marks 2024 MCQ SINGLE
If the sum of all the elements of a \(3\times3\) scalar matrix is 9, then the product of all its elements is:
(A) 0
(B) 9
(C) 27
(D) 729
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Step-by-Step Solution

Let the scalar matrix be represented as:

\[ \begin{bmatrix} k & 0 & 0 \\ 0 & k & 0 \\ 0 & 0 & k \end{bmatrix} \]

The sum of all elements is given as 9:

\[k + 0 + 0 + 0 + k + 0 + 0 + 0 + k = 9\] \[3k = 9\] \[k = 3\]

So, the scalar matrix is:

\[ \begin{bmatrix} 3 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 3 \end{bmatrix} \]

The product of all elements is:

\[3 \times 0 \times 0 \times 0 \times 3 \times 0 \times 0 \times 0 \times 3 = 0\]

Correct Answer: 0

AI Suggestion: Option A

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Pedagogical Audit
Bloom's Analysis: This is an APPLY question because the student needs to apply the concepts of scalar matrices and their properties to solve the problem. They must use the given information to find the scalar value and then calculate the product of all elements.
Knowledge Dimension: CONCEPTUAL
Justification: The question requires understanding the concept of a scalar matrix and how its elements relate to each other. It's not just recalling facts but applying the definition to a specific scenario.
Syllabus Audit: In the context of CBSE Class 12, this is classified as KNOWLEDGE. The question directly tests the understanding and application of matrix types, specifically scalar matrices, which is a standard topic covered in the textbook.

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