Algebra

Matrices & Determinants

Medium

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:

A and B only

A, B and D only

A, B, C and E only

B and D only

✅ Correct Option: 3

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

B: $|AB| = |A||B| = |B||A|$ - true.

C: $adj(A') = (adj\ A)'$ - true.

D: $adj(AB) = (adj\ B)(adj\ A)$ , not $(adj\ A)(adj\ B)$ - false.

E: $|A^n| = |A|^n$ - true.

So A, B, C, E are correct.

2026: 19 May Shift 2

The value of $\begin{vmatrix} \log_3 512 & \log_4 3 \\ \log_3 8 & \log_4 9 \end{vmatrix}$ is:

2025: 21 May Shift 1

The system of equations

$x + ky = 0$

$3x + 5y = 0$

has infinitely many solutions, then k is equal to

2025: 29 May Shift 2

The system of equations

$x - 3y - 8z = -10$

$2x + 5y + \lambda z = 13$

$3x + y - 4z = 0$

has infinite number of solutions if the value of $\lambda$ is equal to:

2022: 7 Aug Shift 2

Match List - I with List - II.

	List - I		List - II
(A)	$A = \begin{bmatrix} 6 & 9 \\ 2 & 3 \end{bmatrix}$ , $B = \begin{bmatrix} 2 & 6 & 0 \\ 7 & 9 & 8 \end{bmatrix}$ , AB will be	(I)	$\begin{bmatrix} 75 & 117 & 72 \\ 35 & 39 & 24 \end{bmatrix}$
(B)	$P = \begin{bmatrix} 8 & 45 & 30 \\ 17 & 19 & 5 \end{bmatrix}$ , $Q = \begin{bmatrix} 67 & 72 & 42 \\ 8 & 30 & 19 \end{bmatrix}$ , P+Q will be	(II)	$\begin{bmatrix} 75 & 117 & 72 \\ 25 & 39 & 24 \end{bmatrix}$
(C)	$\begin{bmatrix} 85 & 42 & 69 \\ 73 & 42 & 50 \end{bmatrix} - \begin{bmatrix} 10 & -75 & -3 \\ 38 & 3 & 26 \end{bmatrix}$	(III)	$\begin{bmatrix} 75 & 117 & 72 \\ 25 & 49 & 24 \end{bmatrix}$
(D)	$2 \begin{bmatrix} 34 & 36 & 30 \\ 12 & 18 & 20 \end{bmatrix} + \begin{bmatrix} 7 & 55 & 12 \\ 1 & 13 & -16 \end{bmatrix}$	(IV)	$\begin{bmatrix} 75 & 127 & 72 \\ 25 & 49 & 24 \end{bmatrix}$

Choose the correct answer from the options given below :

2026: 19 May Shift 2

The value of $\begin{vmatrix} 1 & a & a^2 \\ 1 & b & b^2 \\ 1 & c & c^2 \end{vmatrix}$ is:

2022: 30 Aug Shift 1

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

Solution