Algebra

Matrices & Determinants

Easy

core

The diagonal elements of a skew symmetric matrix are all

Any real number

✅ Correct Option: 1

A matrix is skew-symmetric if $A^T = -A$ , meaning the transpose of the matrix equals the negative of the matrix.

Let $a$ be a diagonal element at position $(i,i)$ .

When taking the transpose of a matrix, diagonal elements remain in the same position.

For a skew-symmetric matrix, applying $A^T = -A$ :

Original diagonal element: $a$

After transpose: $a$

After applying $A^T = -A$ : $-a$

The element must equal its negative:

$a = -a$

$a + a = 0$

$2a = 0$

$a = 0$

For example, consider:

$A = \begin{bmatrix} 0 & 2 & -3 \\ -2 & 0 & 5 \\ 3 & -5 & 0 \end{bmatrix}$

All diagonal elements $(0, 0, 0)$ are zero.

Therefore, all diagonal elements of a skew-symmetric matrix are $0$ .

2022: 7 Aug Shift 2

If A is a square matrix of order 3 and $|A|$ is 2, then value of $|adj(A)|$ is :

2026: 15 May Shift 1

The matrix $\begin{bmatrix} 2 & -1 & 3 \\ \lambda & 0 & 7 \\ -1 & 1 & 4 \end{bmatrix}$ is not invertible for

2025: 19 May Shift 1

If the matrix $M = \begin{bmatrix} 0 & -1 & 3\alpha \\ 1 & \beta & -5 \\ -6 & 5 & 0 \end{bmatrix}$ is skew-symmetric, then

2025: 3 June Shift 2

If $A = \begin{bmatrix} 5 & 2 \\ 4 & 3 \end{bmatrix}$ is a given matrix, then which of the following statements are correct?

(A) $|A| = 7$

(B) minor of $3 = -5$

(D) $adj(A) = \begin{bmatrix} 3 & -2 \\ -4 & 5 \end{bmatrix}$

Choose the correct answer from the options given below:

2026: 29 May Shift 2

If the matrix $A = \begin{bmatrix} -2 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 4 \end{bmatrix}$ is given, then which of the following statements is/are TRUE?

A. A is a square matrix

B. A is a symmetric matrix

C. A is a scalar matrix

D. A is a identity matrix

Choose the correct answer from the options given below:

2025: 27 May Shift 1

If A and B are invertible matrices of the same order, then $(AB)^{-1}$ is equal to

Solution