Q87.Let A be a 2 × 2 matrix with real entries. Let I be the 2 × 2 identity matrix. Denote by tr(A), the sum of diagonal entries of A . Assume that A2 = 1. Statement -1: If A ≠1 and A ≠−1, then det A = −1. Statement −2 : If A ≠1 and A ≠−1, then tr(A) ≠0. (1) Statement −1 is false, Statement −2 is true (2) Statement −1 is true, Statement −2 is true, Statement −2 is a correct explanation for Statement −1 (3) Statement −1 is true, Statement −2 is true; (4) Statement −1 is true, Statement −2 is false. Statement −2 is not a correct explanation for Statement −1
What This Question Tests
This question involves applying properties of 2x2 matrices, specifically the Cayley-Hamilton theorem, to deduce conclusions about the determinant and trace of a matrix given A^2=I and A is not I or -I.
Concepts Tested
Formulas Used
A^2 - tr(A)A + det(A)I = 0 (Cayley-Hamilton for 2x2)
det(A) = ad-bc
tr(A) = a+d
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📋 Question Details
- Chapter
- Matrices
- Topic
- Properties of Matrices
- Year
- 2008
- Shift
- Unknown
- Q Number
- Q87
- Type
- Assertion Reasoning
- NCERT Ref
- Class 12 Mathematics Ch 3: Matrices
More from this Chapter
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