Algebra

Matrices & Determinants

Easy

common

Let $A = [a_{ij}]_{n \times n}$ be a matrix, then match List-I with List-II

List-I List-II
(A) $\vert A\vert = 0$ (I) $A$ is a symmetric matrix
(B) $\vert A\vert \neq 0$ (II) $A$ is a skew-symmetric matrix
(C) $A^T = A$ (III) $A$ is a singular matrix
(D) $A^T = -A$ (IV) $A$ is a non-singular matrix

Choose the correct answer from the options given below:

List-I	List-II
(A) $\vert A\vert = 0$	(I) $A$ is a symmetric matrix
(B) $\vert A\vert \neq 0$	(II) $A$ is a skew-symmetric matrix
(C) $A^T = A$	(III) $A$ is a singular matrix
(D) $A^T = -A$	(IV) $A$ is a non-singular matrix

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

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

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

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

✅ Correct Option: 4

$|A| = 0$ means the determinant is zero, which is the definition of a singular matrix.

$(A) \rightarrow (III)$

$|A| \neq 0$ means the determinant is non-zero, which is the definition of a non-singular matrix.

$(B) \rightarrow (IV)$

$A^T = A$ means the transpose equals the matrix itself, which is the definition of a symmetric matrix.

$(C) \rightarrow (I)$

$A^T = -A$ means the transpose equals the negative of the matrix, which is the definition of a skew-symmetric matrix.

$(D) \rightarrow (II)$

Final matching:

$(A) \rightarrow (III)$

$(B) \rightarrow (IV)$

$(C) \rightarrow (I)$

$(D) \rightarrow (II)$

2025: 14 May Shift 1

Let $A = \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix}$ and $I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$ . If $A^T + A = I$ , then

2026: 15 May Shift 1

Match the LIST-I with LIST-II

	LIST-I (Matrix $A =$ )		LIST-II ( $\vert adj A \vert =$ )
A.	$\begin{bmatrix}2 & 1\\ 1 & 3\end{bmatrix}$	I.	$3$
B.	$\begin{bmatrix}1 & 0\\ 3 & 2\end{bmatrix}$	II.	$2$
C.	$\begin{bmatrix}3 & 0\\ 0 & 1\end{bmatrix}$	III.	$5$
D.	$\begin{bmatrix}0 & -1\\ 4 & 0\end{bmatrix}$	IV.	$4$

Choose the correct answer from the options given below:

2026: 21 May Shift 2

If A is a matrix of order $3 \times 3$ such that $|A| = 4$ and $|k \cdot adj\, A| = 32|A|, k \in \mathbb{R}$ ; then the value of k is:

2026: 29 May Shift 2

If A and B are square matrices of order 3×3 such that |A|=7 and |B|=6, then |B adj A| =

2025: 3 June Shift 1

If the system of equations $2x + 5y = 7, 6x + \lambda y = 28$ is inconsistent, then

2025: 30 May Shift 2

The minimum value of $\begin{vmatrix}2 & 2 & 2 \\ 2 & 2+x & 2 \\ 2 & 2 & 2+x\end{vmatrix}$ , $x \in R$ is

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