Algebra

Matrices & Determinants

Easy

If the matrix $\begin{bmatrix} 0 & -1 & 3x \\ 1 & y & -5 \\ -6 & 5 & 0 \end{bmatrix}$ is skew-symmetric, then

$x = -2, y = 0$

$x = 2, y = 0$

$x = -2, y = 1$

$x = 2, y = -1$

✅ Correct Option: 2

For skew-symmetric matrix, $a_{ij} = -a_{ji}$ and diagonal elements are zero.

$a_{13} = -a_{31}$ gives $3x = 6$ , so $x = 2$ .

$a_{22} = 0$ gives $y = 0$ .

2025: 22 May Shift 2

Match List-I with List-II

Choose the correct answer from the options given below:

2026: 29 May Shift 2

Given that $[a_{ij}]$ is a square matrix of order n.

Match the LIST-I with LIST-II

LIST-I		LIST-II
A.	$A=[a_{ij}]; a_{ij}=\begin{cases}1, & \forall i=j\\0, & \forall i\neq j\end{cases}$	I. Identity Matrix
B.	$B=[a_{ij}]; a_{ij}=\begin{cases}2i-3j, & \forall i=j\\0, & \forall i\neq j\end{cases}$	II. Diagonal Matrix
C.	$C=[a_{ij}]; a_{ij}=\begin{cases}k, & \forall i=j\\0, & \forall i\neq j\end{cases}$	III. Scalar Matrix
D.	$D=[a_{ij}]; a_{ij}=\begin{cases}2i-3j, & \forall i\leq j\\0, & \forall i>j\end{cases}$	IV. Upper Triangular Matrix

Choose the correct answer from the options given below:

2026: 30 May Shift 2

If A and B are symmetric matrices of same order, then which of the following statements are true?

(A) $AB^T+B^TA$ is a skew-symmetric matrix.

(B) $AB^T-BA^T$ is a skew-symmetric matrix

(D) $(AB)^{-1}(BA)^T=A$

Choose the correct answer from the options given below:

2025: 3 June Shift 2

The system of equations

$x + y + z = 4$

$x + 2y + 3z = 12$

$x + 3y + \lambda z = \mu$ has a unique solution if

2022: 6 Aug Shift 2

If A is a matrix of order $m \times n$ and B is another matrix such that $A'B$ and $BA'$ are both defined, then the order of matrix B is

2026: 22 May Shift 2

If $\begin{bmatrix}3 & 1\\2 & 1\end{bmatrix}P\begin{bmatrix}2 & 1\\3 & 2\end{bmatrix}=\begin{bmatrix}1 & -1\\0 & 1\end{bmatrix}$ then matrix P is

If the matrix [0−13x1y−5−650]\begin{bmatrix} 0 & -1 & 3x \\ 1 & y & -5 \\ -6 & 5 & 0 \end{bmatrix}​01−6​−1y5​3x−50​​ is skew-symmetric, then