Algebra

Matrices & Determinants

Medium

applied

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

$\frac{15}{212}$

$\frac{212}{15}$

$\frac{1}{60}$

✅ Correct Option: 4

The question asks for $|(AB)^{-1}|$ , the determinant of the inverse of matrix $(AB)$ .

Using the property of determinants:

$|(AB)^{-1}| = \frac{1}{|AB|}$

$= \frac{1}{|A| \times |B|}$

Matrix $A$ is diagonal, so its determinant equals the product of diagonal elements:

$|A| = 9 \times 16 \times 25$

$= 144 \times 25$

$= 3600$

Matrix $B$ is also diagonal:

$|B| = \frac{1}{5} \times \frac{1}{4} \times \frac{1}{3}$

$= \frac{1}{60}$

Therefore:

$|(AB)^{-1}| = \frac{1}{|A| \times |B|}$

$= \frac{1}{3600 \times \frac{1}{60}}$

$= \frac{1}{\frac{3600}{60}}$

$= \frac{1}{60}$

The answer is $\frac{1}{60}$ .

2025: 15 May Shift 1

Match List-I with List-II

List-I	List-II
(A) If $A$ is a non-singular matrix of order $n$ , then $\vert A(\text{adj} A)\vert$ is equal to	(I) $\vert A\vert ^{n-1}$
(B) If $A$ is a non-singular matrix of order $n$ , then $\vert \text{adj}(\text{adj} A)\vert$ is equal to	(II) $\vert A\vert ^{n-2} A$
(C) If $A$ is a non-singular matrix of order $n$ , then $\text{adj}(\text{adj} A)$ is equal to	(III) $\vert A\vert ^n$
(D) If $A$ is a non-singular matrix of order $n$ , then $\vert (\text{adj} A)\vert$ is equal to	(IV) $\vert A\vert ^{(n-1)^2}$

Choose the correct answer from the options given below:

2023: 30 May Shift 3

Given $A = \begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix}$ and $B = \begin{bmatrix} x & y \\ 1 & 4 \end{bmatrix}$ , If $A = B$ , then $x$ and $y$ are :

2025: 26 May Shift 2

If $\begin{bmatrix} a-b & 0 & 0 \\ 0 & b-c & 0 \\ 0 & 0 & c-2 \end{bmatrix}$ is a scalar matrix such that $a + b + c = 0$ , then, which of the following are TRUE?

(A) $a = 0$

(B) $b = 0$

(D) $c = 1$

Choose the correct answer from the options given below:

2026: 21 May Shift 1

If $\begin{vmatrix} 2+x & y & z \\ x & 2+y & z \\ x & y & 2+z \end{vmatrix} = 0$ , then the value of $x+y+z$ is

2026: 31 May Shift 2

The number of all possible matrices of order $2 \times 2$ with each entry 0 or 1 having their determinants as positive value is :

2025: 16 May Shift 1

If A is a square matrix of order 3 and $|A| = 4$ , then the value of $|2A^T|$ is

Solution

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