Algebra

Matrices & Determinants

Medium

applied

Let $A = [a_{ij}]$ be a square matrix, where $a_{ij} = \begin{cases} 0, & \text{when } i = j \\ 1, & \text{otherwise} \end{cases}$ . If |adj A| = |A|², then which of the following statements are correct?

(A) A is a skew symmetric matrix.

(B) A is a non-singular matrix.

(C) A is a square matrix of order 4.

(D) A is a symmetric matrix.

Choose the correct answer from the options given below:

(A), (B) and (C) only

(B) and (D) only

(A) and (C) only

(B), (C) and (D) only

✅ Correct Option: 2

The matrix A has:

Diagonal elements ( $i = j$ ): $a_{ij} = 0$
All other elements ( $i ≠ j$ ): $a_{ij} = 1$

For order 2: $A = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$

For order 3: $A = \begin{bmatrix} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{bmatrix}$

For order 4: $A = \begin{bmatrix} 0 & 1 & 1 & 1 \\ 1 & 0 & 1 & 1 \\ 1 & 1 & 0 & 1 \\ 1 & 1 & 1 & 0 \end{bmatrix}$

For an $n \times n$ matrix: $|adj(A)| = |A|^{n-1}$

Given condition: $|adj(A)| = |A|^2$

Therefore:

$|A|^{n-1} = |A|^2$

$n - 1 = 2$

$n = 3$

The matrix A is of order 3.

A matrix is skew-symmetric if $A^T = -A$

Since $a_{ij} = a_{ji}$ for all $i, j$ , we have $A^T = A$ , not $-A$

Statement (A) is false.

A matrix is non-singular if $|A| ≠ 0$

For the $3 \times 3$ matrix: $A = \begin{bmatrix} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{bmatrix}$

Expanding along row 1:

$|A| = 0 - 1(0-1) + 1(1-0)$

$|A| = 1 + 1$

$|A| = 2$

Since $|A| = 2 ≠ 0$ , statement (B) is true.

From the given condition, $n = 3$

Statement (C) is false.

A matrix is symmetric if $A^T = A$

Since $a_{ij} = a_{ji}$ for all $i, j$ :

When $i = j$ : both are 0
When $i ≠ j$ : both are 1

Therefore $A^T = A$

Statement (D) is true.

Only statements (B) and (D) are correct.

2026: 30 May Shift 1

The value of $\begin{vmatrix}1 & \log_ba\\\log_ab & 1\end{vmatrix}$ is:

2025: 15 May Shift 1

Let $A = [a_{ij}]_{3 \times 3}$ such that $|A| = -5$ . Then the value of $\det(5A)$ is equal to

2026: 22 May Shift 1

Which of the following are correct?

(A) $A=\begin{bmatrix}-9 & 4 & -3\\-1 & 0 & 4\\2 & 2 & 0\end{bmatrix}$ ; $a_{22}+a_{32}=-9$ where $a_{ij}$ denotes the $(i,j)^{th}$ element of matrix A.

(B) $A=\begin{bmatrix}2 & -2\\-2 & 2\end{bmatrix}$ ; $A^2=pA$ , then $p=4$

(D) $\begin{bmatrix}0 & -5 & 3\\5 & 0 & -7\\-3 & 7 & 0\end{bmatrix}$ is a symmetric matrix

Choose the correct answer from the options given below:

2025: 14 May Shift 2

Let A be any square matrix of order 3 and $B = \begin{bmatrix} 0 & -4 & 2 \\ 4 & 0 & 3 \\ -2 & -3 & 0 \end{bmatrix}$ . Then the matrix $ABA^T$ is a

2026: 26 May Shift 2

The value/s of ' $\lambda$ ' for which the system of equations

$\lambda x + 3y - z = 1$

$x + 2y + z = 2$

$-\lambda x + y + 2z = -1$

has unique solution is/are:

(where $R$ is set of real numbers)

2025: 22 May Shift 1

Let A be a square matrix of order n, then which of the following are TRUE?

(A) $|adj A| = |A|^{n-1}$

(B) $|A. adj A| = |A|^n$

(D) $|KA| = K|A|$

(E) $|A^{-1}| = \frac{1}{|A|}, |A| \neq 0$

Choose the correct answer from the options given below:

Solution

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