Match List-I with List-II
| List-I | List-II | ||
|---|---|---|---|
| (A) | A square matrix P is said to be non-singular if | (I) | \(|P| = 0\) |
| (B) | A square matrix P is said to be singular if | (II) | \(PP^{T}\) is symmetric |
| (C) | If a matrix P is both symmetric and skew-symmetric, then | (III) | \(|P| \ne 0\) |
| (D) | If P is a square matrix, then | (IV) | P is a null matrix |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A)-(III), (B)-(I), (C)-(IV), (D)-(II)
(A) A square matrix P is said to be non-singular if
This matches (III) \(|P| \ne 0\).
By definition, a square matrix is non-singular when its determinant is not zero.
(B) A square matrix P is said to be singular if
This matches (I) \(|P| = 0\).
A square matrix is singular when its determinant is zero.
(C) If a matrix P is both symmetric and skew-symmetric, then
This matches (IV) P is a null matrix.
\(P^{T} = P\) and \(P^{T} = -P\)
\(P = -P\)
\(2P = O\), so \(P = O\)
(D) If P is a square matrix, then
This matches (II) \(PP^{T}\) is symmetric.
By the reversal law for transposes:
\((PP^{T})^{T} = (P^{T})^{T}P^{T} = PP^{T}\)
So the matching is (A)-(III), (B)-(I), (C)-(IV), (D)-(II).
Note: Some List-I items fit more than one List-II entry. In (C), P is the null matrix, so \(|P| = 0\) as well, and (C) also fits (I). The question can still be solved because (A) fits only (III), (B) fits only (I) and (D) fits only (II); Option (3) is the only option containing (A)-(III).