CUET UG 2025 — Mathematics Algebra

A symmetric matrix satisfies $P = P^T$P=PT (flipping across the main diagonal doesn't change it).

A skew-symmetric matrix satisfies $P = -P^T$P=−PT (flipping across the main diagonal gives the negative).

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Since $P$P is both symmetric and skew-symmetric:

$P = P^T$P=PT (symmetric condition)

$P = -P^T$P=−PT (skew-symmetric condition)

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From the symmetric condition, $P^T = P$PT=P.

Substituting into the skew-symmetric condition:

$$P = -P^T$$P=−PT

$$P = -P$$P=−P

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$$P = -P$$P=−P

$$P + P = 0$$P+P=0

$$2P = 0$$2P=0

$$P = 0$$P=0

Therefore, $P$P is a zero matrix (all elements are zero).

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A diagonal matrix includes the zero matrix, but not all diagonal matrices are both symmetric and skew-symmetric.

A square matrix includes the zero matrix, but not all square matrices are both symmetric and skew-symmetric.

The identity matrix is symmetric but not skew-symmetric.

The zero matrix is the only matrix that can be both symmetric and skew-symmetric.

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The answer is: $P$P is a zero matrix.