Algebra

Matrices & Determinants

Hard

core

Let $A = [a_{ij}]$ be a square matrix of order 3 with $|A| = 2$ and let $C = [c_{ij}]$ where $c_{ij} =$ cofactor of $a_{ij}$ in A. Then $|C|$ is equal to:

✅ Correct Option: 2

Given a square matrix $A$ of order 3 with $|A| = 2$ .

Let $C = [c_{ij}]$ where $c_{ij}$ is the cofactor of $a_{ij}$ in $A$ .

For any square matrix $A$ of order $n$ , the relationship between the determinant of $A$ and its cofactor matrix $C$ is:

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

This follows from the property $A \cdot C^T = |A| \cdot I$ , where $C^T$ is the adjugate matrix and $I$ is the identity matrix.

Taking determinants:

$|A \cdot C^T| = ||A| \cdot I|$

$|A| \times |C^T| = |A|^n$

$|A| \times |C| = |A|^n$

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

For the given matrix:

$n = 3$

$|A| = 2$

Substituting into the formula:

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

$|C| = 2^{3-1}$

$|C| = 2^2$

$|C| = 4$

Therefore, $|C| = 4$ .

2026: 15 May Shift 1

For $\Delta = \begin{vmatrix} 1+a & 1 & 1 \\ 1 & 1+b & 1 \\ 1 & 1 & 1+c \end{vmatrix}$ where $a, b, c$ are roots of equation $x^3 + px + q = 0$ , then

Match the LIST-I with LIST-II

	LIST-I		LIST-II
A.	$a + b + c$	I.	$p - q$
B.	$\Delta$	II.	$-q$
C.	$ab + bc + ca$	III.	$0$
D.	$abc$	IV.	$p$

Choose the correct answer from the options given below:

2025: 22 May Shift 1

Let A be a non-singular square matrix of order 3 and $|adj A| = 5$ then $|A|$ is equal to

2025: 13 May Shift 1

If A and B are square matrices of the same order, then (A+B) (A-B) is equal to

2026: 21 May Shift 1

Which of the following matrices are invertible?

(A) $\begin{bmatrix} 2 & 2 \\ 3 & 3 \end{bmatrix}$

(B) $\begin{bmatrix} 2 & 3 \\ 3 & 2 \end{bmatrix}$

(D) $\begin{bmatrix} 3 & 3 \\ 3 & 3 \end{bmatrix}$

Choose the correct answer from the options given below:

2026: 30 May Shift 1

The value of the determinant $\Delta=\begin{vmatrix}\cos\alpha\cos\beta & \cos\alpha\sin\beta & -\sin\alpha\\-\sin\beta & \cos\beta & 0\\\sin\alpha\cos\beta & \sin\alpha\sin\beta & \cos\alpha\end{vmatrix}$ is:

2026: 25 May Shift 1

The value of $\Delta = \begin{vmatrix} 2\sin40^\circ & 2\sin50^\circ \\ -3\cos40^\circ & 3\cos50^\circ \end{vmatrix}$ is:

Let A=[aij]A = [a_{ij}]A=[aij​] be a square matrix of order 3 with ∣A∣=2|A| = 2∣A∣=2 and let C=[cij]C = [c_{ij}]C=[cij​] where cij=c_{ij} =cij​= cofactor of aija_{ij}aij​ in A. Then ∣C∣|C|∣C∣ is equal to:

Solution

Related questions:

Keyboard Shortcuts

CUET Mathematics Past Year Questions (Topic-Wise)

Let $A = [a_{ij}]$ be a square matrix of order 3 with $|A| = 2$ and let $C = [c_{ij}]$ where $c_{ij} =$ cofactor of $a_{ij}$ in A. Then $|C|$ is equal to: