Algebra

Matrices & Determinants

Easy

common

For every square matrix $A$ of order $n\times n$ , which of the following statements are correct ?

(A) $(adjA)A = A(adjA) = |A|I$ , where $I$ is identity matrix of order n.

(B) $|kA|=k^n|A|$ , k is any scalar.

(C) $|adjA|=|A|^n$

(D) $A^2=A$

Choose the correct answer from the options given below:

(A) and (B) only

(B) and (C) only

(A) and (C) only

(B) and (D) only

✅ Correct Option: 1

(A) is the standard adjoint identity, and (B) holds because each of the $n$ rows contributes one factor of $k$ .

(C) is false: the correct result is $|adjA|=|A|^{n-1}$ . (D) holds only for idempotent matrices, not every square matrix. Hence only (A) and (B) are correct.

2025: 26 May Shift 2

If $y = -4$ is a root of $\begin{vmatrix} y & 2 & 3 \\ 1 & y & 1 \\ 3 & 2 & y \end{vmatrix} = 0$ , then the product of the other two roots is

2025: 22 May Shift 2

Consider the matrices $A = \begin{bmatrix} 9 & 0 & 0 \\ 0 & 16 & 0 \\ 0 & 0 & 25 \end{bmatrix}$ and $B = \begin{bmatrix} \frac{1}{5} & 0 & 0 \\ 0 & \frac{1}{4} & 0 \\ 0 & 0 & \frac{1}{3} \end{bmatrix}$ . The value of $|(AB)^{-1}|$ is

2025: 3 June Shift 1

Let P and Q be any two invertible matrices of the same order. Then

Match List-I with List-II

List-I	List-II
Matrix	Equivalent matrix
(A) $(P Q)^{-1}$	(I) $Q^{-1}P$
(B) $(P^{-1}Q)^{-1}$	(II) $Q P^{-1}$
(C) $(P Q^{-1})^{-1}$	(III) $Q^{-1}P^{-1}$
(D) $(P^{-1}Q^{-1})^{-1}$	(IV) Q P

Choose the correct answer from the options given below:

2025: 26 May Shift 2

If a matrix $A = \begin{bmatrix} 5 & -8 \\ -3 & 5 \end{bmatrix}$ then which of the following is / are TRUE?

(A) $|A| = 1$

(B) $A$ is a singular matrix.

(D) $AI = \begin{bmatrix} 5 & 0 \\ 0 & 5 \end{bmatrix}$ $I$ is an identity matrix of order 2.

Choose the correct answer from the options given below:

2025: 13 May Shift 1

If $A = [a_{ij}]_{3×3}$ where $a_{ij} = \begin{cases} (-1)^{i+j} - 1, & i = j \\ (-1)^{i+j}, & i \neq j \end{cases}$ then the value of $A + A^T$ is:

2026: 30 May Shift 2

If A and B are invertible matrices of the same order, then

Match List-I with List-II

List-I (Determinant)	List-II (Equal to)
(A) $\left\vert A^{-1}B \right\vert$	(I) $\dfrac{1}{\left\vert A \right\vert \left\vert B \right\vert}$
(B) $\left\vert AB^{-1} \right\vert$	(II) $\dfrac{\left\vert B \right\vert}{\left\vert A \right\vert}$
(C) $\left\vert A^{-1}B^{-1} \right\vert$	(III) $\left\vert A \right\vert \left\vert B \right\vert$
(D) $\left\vert AB \right\vert$	(IV) $\dfrac{\left\vert A \right\vert}{\left\vert B \right\vert}$

Choose the correct answer from the options given below:

Solution

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