Algebra

Matrices & Determinants

Medium

core

Let $A = [a_{ij}]_{2 \times 3}$ and $B = [b_{ij}]_{3 \times 2}$ , then $|5AB|$ is equal to

$5^2 .|A|.|B|$

$5^3 .|A|.|B|$

$5^2|AB|$

$5^3 |AB|$

✅ Correct Option: 3

Given:

Matrix $A$ is a $2 \times 3$ matrix (2 rows, 3 columns)
Matrix $B$ is a $3 \times 2$ matrix (3 rows, 2 columns)

When multiplying $A \times B$ :

Result $= (2 \times 3) \times (3 \times 2) = 2 \times 2$ matrix

So $AB$ is a square matrix of order $2 \times 2$ .

Determinants (the $| |$ symbol) only exist for square matrices.

$A$ is $2 \times 3$ → not square → no determinant exists

$B$ is $3 \times 2$ → not square → no determinant exists

$AB$ is $2 \times 2$ → square → determinant exists

When a scalar $k$ multiplies an $n \times n$ matrix $M$ :

$|kM| = k^n |M|$

where $n$ is the order of the square matrix.

For this problem:

$AB$ is a $2 \times 2$ matrix, so $n = 2$

Scalar multiplier $k = 5$

Applying the formula:

$|5AB| = 5^2 |AB|$

$|5AB| = 25|AB|$

Therefore, $|5AB| = 5^2|AB|$ .

option (A) does not satisfy the result; it is not given as correct in the NTA key, it is due to the fact that determinants $|A|$ and $|B|$ are not defined for non-square matrices.

2022: 4 Aug Shift 1

If A and B are square matrices of same order n, then identify correct statements from the statements given below:

A. $|adj\ A| = |A|^{n-1}$

B. $|A \cdot B| = |B| \cdot |A|$

C. $adj\ A' = (adj\ A)'$

D. $adj\ AB = (adj\ A) \cdot (adj\ B)$

E. $|A^n| = |A|^n$

Choose the correct answer from the options given below:

2022: 23 Aug Shift 1

Choose the correct statement

A. If any two rows or any two columns are identical or proportional, then value of determinant is Zero.

B. Minor of an element $a_{ij}$ of the determinant of matrix A is the determinant obtained by deleting $i^{th}$ row and $j^{th}$ column

C. If $A = \begin{bmatrix} 1 & 5 \\ 6 & 7 \end{bmatrix}$ , then A is Skew-symmetric matrix

D. If $A = \begin{bmatrix} 1 & 2 \\ 4 & 5 \end{bmatrix}$ , then $(A + A')$ is Symmetric matrix

E. If $|A| = 0$ , then A is non-singular matrix

2025: 13 May Shift 2

For the system $\begin{bmatrix} 2 & -3 \\ -4 & 6 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 5 \\ -10 \end{bmatrix}$ which of the following statements are correct?

(A) The system has no solution.

(B) The system is consistent.

(D) It has a unique solution.

Choose the correct answer from the options given below:

2026: 31 May Shift 2

Let $A = [a_{ij}]_{3\times3}$ be a matrix such that $a_{ij} = \dfrac{i-j}{i+j}$ , then which of the following statements is/are TRUE?

A. A is a symmetric matrix

B. A is a skew symmetric matrix

C. $|A| = 0$

D. $|A| \neq 0$

Choose the correct answer from the options given below:

2025: 30 May Shift 1

If $A = \begin{bmatrix} 2 & -2 & 1 \\ 0 & 4 & 2 \end{bmatrix}$ and $B = \begin{bmatrix} 1 & -2 & 7 \\ 2 & 0 & 6 \end{bmatrix}$ are two matrices such that $3A - 2B + 4C = 0$ , then matrix $C$ is equal to:

2022: 17 Aug Shift 2

If $A$ is a skew-symmetric matrix and $n$ is an odd positive integer, then $A^n$ is :

Let A=[aij]2×3A = [a_{ij}]_{2 \times 3}A=[aij​]2×3​ and B=[bij]3×2B = [b_{ij}]_{3 \times 2}B=[bij​]3×2​, then ∣5AB∣|5AB|∣5AB∣ is equal to

Solution

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CUET Mathematics Past Year Questions (Topic-Wise)

Let $A = [a_{ij}]_{2 \times 3}$ and $B = [b_{ij}]_{3 \times 2}$ , then $|5AB|$ is equal to