Let \(A\) be a \(2 \times 2\) matrix Statement-1: \(\text{adj}(\text{adj } A) = A\) Statement-2: \(|\text{adj } A| = |A|\)

(B) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1

(A) 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 false

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

JEE Main · 2009 · 12 Apr 2009 · Offline · Matrices

Let

A

be a

2 \times 2

matrix

Statement-1:

adj (adj A) = A

Statement-2:

∣ adj A ∣ = ∣ A ∣

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

Answer: (B)

For an

n \times n

matrix

∣ adj A ∣ = ∣ A ∣^{n - 1}

, which for a

2 \times 2

matrix is just

∣ A ∣

, so the reason statement is true. Also

adj (adj A) = ∣ A ∣^{n - 2} A

, and for

n = 2

the exponent is zero, giving

adj (adj A) = A

, so the assertion is true as well. But the assertion rests on the exponent

n - 2

being zero rather than on the determinant identity, so the second statement, though true, is not the correct explanation of the first.

JEE Main · 2009 · 12 Apr 2009 · Offline · Single correct option