Matrices & Determinants
Matrices and Determinants
Grade Class 12
Question:
Let A be a 2 x 2 matrix with non-zero entries and let A<sup>2</sup> = I, where I is 2 x 2 identity matrix. Define Tr(A) = sum of diagonal elements of A and |A| = determinant of matrix A.<br>Statement-1 : Tr(A) = 0.<br>Statement-2 : |A| = 1.<br>(A) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for statement-1.<br>(B) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for statement-1.<br>(C) Statement-1 is true, Statement-2 is false.<br>(D) Statement-1 is false, Statement-2 is true.
Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for statement-1.
Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for statement-1.
Statement-1 is true, Statement-2 is false.
Statement-1 is false, Statement-2 is true.
Step-by-Step Solution
Key Concept: For a 2x2 matrix A with A^2 = I, the characteristic equation is A^2 - Tr(A)A + |A|I = 0. Since A^2 = I, we have I - Tr(A)A + |A|I = 0, which implies Tr(A)A = (1+|A|)I. If Tr(A) is not 0, then A is a scalar multiple of I, which contradicts the condition that all entries are non-zero. Thus Tr(A) = 0. Then 1+|A| = 0, so |A| = -1.
Let A = [[a, b], [c, d]]. A^2 = I implies the characteristic equation is A^2 - Tr(A)A + |A|I = 0. Since A^2 = I, we have I - Tr(A)A + |A|I = 0, or Tr(A)A = (1+|A|)I. If Tr(A) != 0, then A = ((1+|A|)/Tr(A))I, which means A is a diagonal matrix with equal diagonal elements and zero off-diagonal elements, contradicting the non-zero entries condition. Thus Tr(A) = 0. Then 1+|A| = 0, so |A| = -1. Statement-1 is true, Statement-2 is false.
Correct Answer: 3