Algebra

Matrices & Determinants

Medium

core

Match List-I with List-II

List-I List-II
(A) A square matrix $P$ is said to be non-singular if (I) $\vert P\vert = 0$
(B) A square matrix $P$ is said to be singular if (II) $P P^T$ is symmetric
(C) If a matrix $P$ is both symmetric and skew-symmetric, then (III) $\vert P\vert \neq 0$
(D) If $P$ is a square matrix, then (IV) $P$ is a null matrix

Choose the correct answer from the options given below:

List-I	List-II
(A) A square matrix $P$ is said to be non-singular if	(I) $\vert P\vert = 0$
(B) A square matrix $P$ is said to be singular if	(II) $P P^T$ is symmetric
(C) If a matrix $P$ is both symmetric and skew-symmetric, then	(III) $\vert P\vert \neq 0$
(D) If $P$ is a square matrix, then	(IV) $P$ is a null matrix

(A) - (I), (B) - (II), (C) - (III), (D) - (IV)

(A) - (II), (B) - (III), (C) - (I), (D) - (IV)

(A) - (III), (B) - (I), (C) - (IV), (D) - (II)

(A) - (I), (B) - (IV), (C) - (II), (D) - (III)

✅ Correct Option: 3

A square matrix $P$ is non-singular when its determinant is not equal to zero.

$(A) \rightarrow (III) \quad |P| \neq 0$

A square matrix $P$ is singular when its determinant equals zero.

$(B) \rightarrow (I) \quad |P| = 0$

Symmetric means $P^T = P$ and skew-symmetric means $P^T = -P$ .

If both are true:

$P = P^T = -P$

$P = -P$

$2P = O$

$P = O \text{ (null matrix)}$

$(C) \rightarrow (IV) \quad P \text{ is a null matrix}$

Let $Q = PP^T$

$Q^T = (PP^T)^T$

Using $(XY)^T = Y^T X^T$ :

$Q^T = (P^T)^T \cdot P^T$

$= P \cdot P^T$

$= Q$

Since $Q^T = Q$ , the matrix $PP^T$ is always symmetric.

$(D) \rightarrow (II) \quad PP^T \text{ is symmetric}$

List-I	List-II
(A)	(III)
(B)	(I)
(C)	(IV)
(D)	(II)

The correct answer is Option 3.

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(B) Unit matrix

(D) Skew-symmetric matrix

Choose the correct answer from the options given below:

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Solution

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