Matrices — JEE Main · 2008 · 27 Apr 2008 · Offline Question with Solution

Let A be a

2 \times 2

matrix with real entries. Let I be the

2 \times 2

identity matrix. Denote by

tr (A)

, the sum of diagonal entries of

A

. Assume that

A^{2} = I

.
Statement -1: If

A \neq = I

and

A \neq = - I

, then

det A = - 1

.
Statement -2 : If

A \neq = I

and

A \neq = - I

, then

tr (A) \neq = 0

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

Answer: (D)

Solution

Write

A = (a c b d)

. The condition

A^{2} = I

forces

(a + d) b = 0

and

(a + d) c = 0

together with

a^{2} + b c = 1

; if

a + d

were non-zero then

b = c = 0

and

A

would be

\pm I

, which is excluded, so

a + d = 0

. Then

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

, so the first statement holds. But

a + d = 0

means the trace is zero, so the second statement, which claims a non-zero trace, is false.

JEE Main · 2008 · 27 Apr 2008 · Offline · Single correct option