If a matrix P is both symmetric and skew-symmetric, then Option 1: P is a diagonal matrix Option 2: P is a square matrix Option 3: P is a zero matrix Option 4: P is an identity matrix

Answer: P is a zero matrix. Solution: A symmetric matrix satisfies P = P^T (flipping across the main diagonal doesn't change it). A skew-symmetric matrix satisfies P = -P^T (flipping across the main diagonal gives the negative). Since P is both symmetric and skew-symmetric: P = P^T (symmetric condition) P = -P^T (skew-symmetric condition) From the symmetric condition, P^T = P. Substituting into the skew-symmetric condition: P = -P^T P = -P P = -P P + P = 0 2P = 0 P = 0 Therefore, P is a zero matrix (all elements are zero). 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. The answer is: P is a zero matrix.

Algebra

Matrices & Determinants

Easy

core

If a matrix P is both symmetric and skew-symmetric, then

P is a diagonal matrix

P is a square matrix

P is a zero matrix

P is an identity matrix

✅ Correct Option: 3

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

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

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

$P = P^T$ (symmetric condition)

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

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

Substituting into the skew-symmetric condition:

$P = -P^T$

$P = -P$

$P + P = 0$

$2P = 0$

$P = 0$

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

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.

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

2025: 30 May Shift 1

Let $A$ be a non singular matrix of order $n \times n$ , Then $|\text{adj }(3A)|$ is equal to:

2025: 30 May Shift 1

If $A$ and $B$ are invertible matrices of order $3$ then match List-I with List-II

List-I	List-II
(A) $\text{adj}(A)$	(I) $B^{-1} A^{-1}$
(B) $(AB)^{-1}$	(II) $\vert A\vert ^{-1}$
(C) $\vert A^{-1}\vert$	(III) $\vert A\vert ^2$
(D) $\vert \text{adj} A\vert$	(IV) $\vert A\vert A^{-1}$

Choose the correct answer from the options given below:

2026: 31 May Shift 2

If $\begin{bmatrix} 0 & x \\ 2 & 1 \end{bmatrix} + 2\begin{bmatrix} 3 & 1 \\ y & 0 \end{bmatrix} = \begin{bmatrix} 6 & 5 \\ 8 & 1 \end{bmatrix}$ , then the value of $2x+3y$ is :

2025: 30 May Shift 1

Let $A = \begin{bmatrix} 2 & -1 & 3 \\ 1 & 2 & -1 \\ 4 & 1 & 2 \end{bmatrix}$ . $M_{ij}$ and $A_{ij}$ respectively denote the minor and cofactor of an element $a_{ij}$ of matrix $A = [a_{ij}]$

(A) $M_{23} = 6$

(B) $A_{22} = -8$

(D) $M_{32} = -5$

Choose the correct answer from the options given below:

2025: 30 May Shift 2

Given a matrix A of order 3x3. If |A|=3 then the value of |A(adj A)| is:

2026: 29 May Shift 1

Let $A=\begin{bmatrix}2 & 2 & 2 \\ 0 & 2 & 2 \\ 0 & 0 & 2\end{bmatrix}$ , then $A^6$ is equal to:

If a matrix P is both symmetric and skew-symmetric, then

Solution

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