Algebra

Matrices & Determinants

Medium

common

If $a,b,c$ are the roots of $x^3+px+q=0$ , then the value of $\begin{vmatrix} a & b & c \\ b & c & a \\ c & a & b \end{vmatrix}$ is equal to

p-q

$p^2+q^2-pq$

✅ Correct Option: 1

This circulant determinant equals $-(a^3+b^3+c^3-3abc)=-(a+b+c)(a^2+b^2+c^2-ab-bc-ca)$ .

Since the cubic has no $x^2$ term, $a+b+c=0$ , so the whole product is zero.

2025: 14 May Shift 1

Match List-I with List-II

List-I	List-II
(Matrix)	(Inverse of the Matrix)
(A) $\begin{pmatrix} 1 & 7 \\ 4 & -2 \end{pmatrix}$	(I) $\begin{pmatrix} 2/15 & 1/10 \\ -1/15 & 1/5 \end{pmatrix}$
(B) $\begin{pmatrix} 6 & -3 \\ 2 & 4 \end{pmatrix}$	(II) $\begin{pmatrix} 1/5 & -2/15 \\ -1/10 & 7/30 \end{pmatrix}$
(C) $\begin{pmatrix} 5 & 2 \\ -5 & 4 \end{pmatrix}$	(III) $\begin{pmatrix} 1/15 & 7/30 \\ 2/15 & -1/30 \end{pmatrix}$
(D) $\begin{pmatrix} 7 & 4 \\ 3 & 6 \end{pmatrix}$	(IV) $\begin{pmatrix} 2/15 & -1/15 \\ 1/6 & 1/6 \end{pmatrix}$

Choose the correct answer from the options given below:

2025: 3 June Shift 1

The value of k for which the system of equations

$x + y + z = 1$

$x - ky + z = 1$

$x - y + z = 1$

has more than one solutions is

2025: 21 May Shift 2

Let A be a 3 × 7 matrix, then each column of A contains:

2026: 18 May Shift 2

If $A=\begin{bmatrix} 1 & 0 \\ -1 & 5 \end{bmatrix}$ and I is indentity matrix of order 2, then which of following statements are TRUE?

A. $A^2-6A-mI=0$ if $m=-5$ .

B. $A^2-6A+mI=0$ if $m=-5$ .

C. $A$ is a non-singular matrix.

D. $A^2=A$

Choose the correct answer from the options given below:

2025: 19 May Shift 1

If the system of equations $2x + 3y = 10$ , $x + ky = 4$ has a unique solution, then

2026: 30 May Shift 1

If $A$ is a square matrix of order 3 and $|A|=6$ , then the value of $|2A^T|$ is:

If a,b,ca,b,ca,b,c are the roots of x3+px+q=0x^3+px+q=0x3+px+q=0, then the value of ∣abcbcacab∣\begin{vmatrix} a & b & c \\ b & c & a \\ c & a & b \end{vmatrix}​abc​bca​cab​​ is equal to