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) $|P| ≠ 0$ |
|
(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)
|
List-I |
List-II |
|
(A) A square matrix P is said to be non-singular if |
(III) $|P| ≠ 0$ |
|
(B) A square matrix P is said to be singular if |
(I) $|P| = 0$ |
|
(C) If a matrix P is both symmetric and skew-symmetric, then |
(IV) P is a null matrix |
|
(D) If P is a square matrix, then |
(II) $PP^T$ is symmetric |
(A) A matrix is said to be non-singular if its determinant is nonzero, i.e. $|P| \ne 0$. This means the matrix has an inverse. → (III)
(B) A matrix is said to be singular if its determinant is zero, i.e. $|P| = 0$. This means the matrix does not have an inverse. → (I)
(C) If a matrix is both symmetric ($P^T = P$) and skew-symmetric ($P^T = -P$), then combining gives $P = -P \Rightarrow 2P = 0 \Rightarrow P = 0$. Hence, it must be a null matrix. → (IV)
(D) For any square matrix $P$, the product $PP^T$ is always symmetric, because $(PP^T)^T = (P^T)^T P^T = PP^T$. → (II)
Final Matching:
(A)–(III), (B)–(I), (C)–(IV), (D)–(II)