Class CBSE Class 12 Mathematics Matrices and Determinants Q #888
KNOWLEDGE BASED
APPLY
3 Marks 2023 SA
If A=[(1, 2, 3), (3, -2, 1), (4, 2, 1)], then show that A³ - 23A - 40I = O.
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Step-by-Step Solution

Step 1: Calculate A²

A² = A * A = [(1, 2, 3), (3, -2, 1), (4, 2, 1)] * [(1, 2, 3), (3, -2, 1), (4, 2, 1)]

A² = [ (1*1 + 2*3 + 3*4, 1*2 + 2*(-2) + 3*2, 1*3 + 2*1 + 3*1), (3*1 + (-2)*3 + 1*4, 3*2 + (-2)*(-2) + 1*2, 3*3 + (-2)*1 + 1*1), (4*1 + 2*3 + 1*4, 4*2 + 2*(-2) + 1*2, 4*3 + 2*1 + 1*1) ]

A² = [(1+6+12, 2-4+6, 3+2+3), (3-6+4, 6+4+2, 9-2+1), (4+6+4, 8-4+2, 12+2+1)]

A² = [(19, 4, 8), (1, 12, 8), (14, 6, 15)]

Step 2: Calculate A³

A³ = A² * A = [(19, 4, 8), (1, 12, 8), (14, 6, 15)] * [(1, 2, 3), (3, -2, 1), (4, 2, 1)]

A³ = [ (19*1 + 4*3 + 8*4, 19*2 + 4*(-2) + 8*2, 19*3 + 4*1 + 8*1), (1*1 + 12*3 + 8*4, 1*2 + 12*(-2) + 8*2, 1*3 + 12*1 + 8*1), (14*1 + 6*3 + 15*4, 14*2 + 6*(-2) + 15*2, 14*3 + 6*1 + 15*1) ]

A³ = [(19+12+32, 38-8+16, 57+4+8), (1+36+32, 2-24+16, 3+12+8), (14+18+60, 28-12+30, 42+6+15)]

A³ = [(63, 46, 69), (69, -6, 23), (92, 46, 63)]

Step 3: Calculate 23A

23A = 23 * [(1, 2, 3), (3, -2, 1), (4, 2, 1)] = [(23, 46, 69), (69, -46, 23), (92, 46, 23)]

Step 4: Calculate 40I

40I = 40 * [(1, 0, 0), (0, 1, 0), (0, 0, 1)] = [(40, 0, 0), (0, 40, 0), (0, 0, 40)]

Step 5: Calculate A³ - 23A - 40I

A³ - 23A - 40I = [(63, 46, 69), (69, -6, 23), (92, 46, 63)] - [(23, 46, 69), (69, -46, 23), (92, 46, 23)] - [(40, 0, 0), (0, 40, 0), (0, 0, 40)]

A³ - 23A - 40I = [(63-23-40, 46-46-0, 69-69-0), (69-69-0, -6+46-40, 23-23-0), (92-92-0, 46-46-0, 63-23-40)]

A³ - 23A - 40I = [(0, 0, 0), (0, 0, 0), (0, 0, 0)]

Step 6: Conclusion

A³ - 23A - 40I = O

Correct Answer: O

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Pedagogical Audit
Bloom's Analysis: This is an APPLY question because the student needs to apply the knowledge of matrix multiplication and substitution to verify the given equation.
Knowledge Dimension: PROCEDURAL
Justification: The question requires the student to follow a specific procedure of matrix multiplication and substitution to arrive at the solution.
Syllabus Audit: In the context of CBSE Class 12, this is classified as KNOWLEDGE. The question directly tests the student's understanding and application of matrix operations as covered in the textbook.

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