Question

Let A be a 2 × 2 real matrix with entries from {0, 1} and |A| ≠ 0. Consider the following two statements:

(P) If A¹I₂, then |A| = –1

(Q) If |A| = 1, then tr(A) = 2,

where I₂ denotes 2 × 2 identity matrix and tr(A) denotes the sum of the diagonal entries of A. Then ______.

Options

• Both (P) and (Q) are false

• (P) is true and (Q) is false

• Both (P) and (Q) are true

• (P) is false and (Q) is true

Answer 1

Let A be a 2 × 2 real matrix with entries from {0, 1} and |A| ≠ 0. Consider the following two statements:

(P) If A¹I₂, then |A| = –1

(Q) If |A| = 1, then tr(A) = 2,

where I₂ denotes 2 × 2 identity matrix and tr(A) denotes the sum of the diagonal entries of A. Then (P) is false and (Q) is true.

Explanation:

P: A = `[(1, 0),(1, 0)] ≠ I_2`

|A| = 1, but |A| = –1  ...(Given)

⇒ P is false

Q: A = `[(1, 1),(0, 1)]` or `[(1, 0),(1, 1)]`

= `[(1, 0),(0, 1)]`

Value of all the above matrices is 1 and trace of A = 2 in all cases.

⇒ Q is true