Algebra

Matrices & Determinants

Medium

core

Let $A=[a_{ij}]_{3\times3}$ , where $a_{ij}=i-j$ , then which of the following statements is/are TRUE ?

A. A is diagonal matrix

B. A is square matrix

C. A is symmetric matrix

D. A is skew symmetric matrix

Choose the correct answer from the options given below:

A and B only

B and D only

B and C only

A and C only

✅ Correct Option: 2

$A=\begin{bmatrix}0&-1&-2\\1&0&-1\\2&1&0\end{bmatrix}$ is $3\times3$ , hence square (B true). Since $a_{ji}=j-i=-a_{ij}$ and the diagonal is zero, it is skew symmetric (D true). Non-zero off-diagonal entries rule out diagonal, and $a_{12}\ne a_{21}$ rules out symmetric.

2026: 31 May Shift 2

Let $A = \begin{bmatrix} 1 & -2 & 3 \\ 1 & 2 & 1 \\ \lambda & 2 & -3 \end{bmatrix}$ . If $A^{-1}$ does not exist, then $\lambda =$

2026: 15 May Shift 1

If $A$ is a square matrix of order 3 such that $|A|=3$ , then the value of $|adj(adjA)|$ is equal to

2025: 22 May Shift 2

Match List-I with List-II

List-I	List-II
Matrix Product	Order of resultant matrix
---	---
(A) $[a_{ij}]_{2 \times 3} \times [b_{ij}]_{3 \times 4}$	(I) $2 \times 4$
(B) $[a_{ij}]_{2 \times 1} \times [b_{ij}]_{1 \times 3}$	(II) Not possible
(C) $[a_{ij}]_{3 \times 2} \times [b_{ij}]_{3 \times 2}$	(III) $3 \times 3$
(D) $[a_{ij}]_{3 \times 3} \times [b_{ij}]_{3 \times 3}$	(IV) $2 \times 3$

Choose the correct answer from the options given below:

2022: 30 Aug Shift 1

If $2 \begin{bmatrix} a & d \\ b & c \end{bmatrix} + 3 \begin{bmatrix} 1 & -1 \\ 0 & 2 \end{bmatrix} = 3 \begin{bmatrix} 3 & 5 \\ 4 & 6 \end{bmatrix}$ , then the value of $|a + b - c - d|$ is

2025: 13 May Shift 1

If $M = \begin{bmatrix} 2 \\ -1 \\ 3 \end{bmatrix}$ and $N = [7 \quad 1 \quad -4]$ , then $(MN)^T$ will be equal to:

2026: 29 May Shift 2

With reference to the matrix $A=\begin{bmatrix}5&0&0\\0&5&0\\0&0&-5\end{bmatrix}$ , which of the following are TRUE ?

A. |A|=125

B. A is a scalar matrix.

C. A is a diagonal matrix.

D. |A|=-125

Choose the correct answer from the options given below:

Solution

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