Algebra

Matrices & Determinants

Medium

common

If A is square matrix of order 3 × 3 and |adj A| = 64, then the value of |5A| is

100

± 1000

320

± 320

✅ Correct Option: 2

For any $n \times n$ matrix $A$ :

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

For any $n \times n$ matrix $A$ and scalar $k$ :

$|kA| = k^n \times |A|$

Given that $A$ is a $3 \times 3$ matrix and $|adj A| = 64$ .

Using the formula for determinant of adjoint:

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

$|adj A| = |A|^{3-1}$

$|adj A| = |A|^2$

$64 = |A|^2$

$|A| = \pm 8$

Using the formula for determinant of scalar multiple with $k = 5$ and $n = 3$ :

$|5A| = 5^n \times |A|$

$|5A| = 5^3 \times |A|$

$|5A| = 125 \times (\pm 8)$

$|5A| = \pm 1000$

Therefore, the value of $|5A|$ is $\pm 1000$ .

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Match List-I with List-II

List-I	List-II
A is given matrix	$\left\vert \text{adj}(\text{adj}A) \right\vert$
(A) Order of A $= 2 \times 2$ and $\left\vert A \right\vert = 3$	(I) $3$
(B) Order of A $= 3 \times 3$ and $\left\vert A \right\vert = 2$	(II) $1$
(C) Order of A $= 4 \times 4$ and $\left\vert A \right\vert = 1$	(III) $16$
(D) Order of A $= 2 \times 2$ and $\left\vert A \right\vert = 4$	(IV) $4$

Choose the correct answer from the options given below:

2025: 3 June Shift 1

If $A = \begin{bmatrix} 1 & 2 \\ 4 & 5 \end{bmatrix}$ , then

Match List-I with List-II

Choose the correct answer from the options given below:

2022: 30 Aug Shift 1

The equations in terms of $x$ and $y$ are:

Solution