Question

Let A be a 2 × 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 ______.

पर्याय

• –1

• 2

• 1

• `-sqrt(2)`

Answer 1

Let A be a 2 × 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 2.

Explanation:

Given relation det ((A + I)(adj(A) + I)) = 4, det (A) = –1,

Then, adj A = –A^–1.

|(A + I)A^–1 + I| = 4

|–I + A – A^–1 + I| = 4

|A – A^–1| = 4

Let A = `[(a, b),(c, d)]` then A^–1 = `[(-d, b),(c, -a)]`

|A – A^–1| = `[(a + d, 0),(0, d + a)]` = 4

(a + d)² = 4

`\implies` a + d = ±2

`\implies` |a + d| = 2