Let A be a $2 \times 2$ matrix with $\operatorname{det}(\mathrm{A})=-1$ and $\operatorname{det} ((\mathrm{A}+\mathrm{I})(\operatorname{Adj}(\mathrm{A})+\mathrm{I}))=4$. Then the sum of the diagonal elements of A can be :

JEE Main · 2022 · 26 Jul 2022 · Shift 1 · Matrices

Let A be a

2 \times 2

matrix with

det (A) = - 1

and

det ((A + I) (Adj (A) + I)) = 4

. Then the sum of the diagonal elements of A can be :

Answer: (B)

Let

A = (a c b d)

with

det (A) = a d - b c = - 1

.

Using the property

det ((A + I) (Adj (A) + I)) = det (A + I) \cdot det (Adj (A) + I) = 4

.

For a

2 \times 2

matrix,

Adj (A) = (d - c - b a)

det (A + I) = (a + 1) (d + 1) - b c = a d - b c + a + d + 1 = - 1 + a + d + 1 = a + d

det (Adj (A) + I) = (d + 1) (a + 1) - b c = a + d

.

Thus

(a + d)^{2} = 4

, giving

a + d = \pm 2

.

Both values are possible, and option (B) gives

a + d = 2

JEE Main · 2022 · 26 Jul 2022 · Shift 1 · Single correct option