If … where a, b, c, d are integers, then which of the following are true? (A) a + d = 0 (B) b + d = 3 (C) b + d = 1 (D) c + d = 2 Choose the correct…

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

$$(A) \rightarrow (III) \quad |P| \neq 0$$(A)→(III)∣P∣=0

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

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

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

If both are true:

$$P = P^T = -P$$P=PT=−P

$$P = -P$$P=−P

$$2P = O$$2P=O

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

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

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Let $Q = PP^T$Q=PPT

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

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

$$Q^T = (P^T)^T \cdot P^T$$QT=(PT)T⋅PT

$$= P \cdot P^T$$=P⋅PT

$$= Q$$=Q

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

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

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

The correct answer is Option 3.