Let A be a $2 \times 2$ matrix with non-zero entries and let $A^{2}=I$, where I is $2 \times 2$ identity matrix. Define $\operatorname{Tr}(A)=$ sum of diagonal elements of A and $|A|=$ determinant of matrix A. Statement-1: $\operatorname{Tr}(A)=0$ Statement-2: $|A|=1$

(B) Statement-1 is true, Statement-2 is false

(A) Statement-1 is true, Statement-2 is true; Statement-2 is not the correct explanation for Statement-1

(C) Statement-1 is false, Statement-2 is true

(D) Statement-1 is true, Statement-2 is true; Statement-2 is the correct explanation for Statement-1

Matrices — JEE Main · 2010 · 11 Apr 2010 · Offline Question with Solution

Let A be a

2 \times 2

matrix with non-zero entries and let

A^{2} = I

, where I is

2 \times 2

identity matrix. Define

Tr (A) =

sum of diagonal elements of A and

∣ A ∣ =

determinant of matrix A.

Statement-1:

Tr (A) = 0

Statement-2:

∣ A ∣ = 1

(A)Statement-1 is true, Statement-2 is true; Statement-2 is not the correct explanation for Statement-1
(B)Statement-1 is true, Statement-2 is false✓ Correct
(C)Statement-1 is false, Statement-2 is true
(D)Statement-1 is true, Statement-2 is true; Statement-2 is the correct explanation for Statement-1

Answer: (B)

Let

A = (a c b d)

with every entry non-zero and

A^{2} = I

. The off-diagonal entries of

A^{2}

give

b (a + d) = 0

and

c (a + d) = 0

; since

b, c \neq = 0

, we must have

a + d = 0

, so the trace

a + d = 0

and Statement-1 is true. For the determinant, the diagonal entries give

a^{2} + b c = 1

, so

∣ A ∣ = a d - b c = - a^{2} - b c = - (a^{2} + b c) = - 1

, not

+ 1

. Statement-2 is therefore false, and the correct reading is that Statement-1 is true while Statement-2 is false.

JEE Main · 2010 · 11 Apr 2010 · Offline · Single correct option