Algebra

Matrices & Determinants

Medium

core

If $\Delta = \begin{vmatrix} 1+a & 1 & 1 \\ 1 & 1+b & 1 \\ 1 & 1 & 1+c \end{vmatrix}$ , Then

Match List-I with List-II

List-I List-II
Values of $a, b$ & $c$ value of $\Delta$
(A). $a=b=c=1$ (I). 2
(B). $a=-1,b=1,c=1$ (II). 0
(C). $a=-1,b=-1,c=1$ (III). -2
(D). $a=-1,b=-1,c=-1$ (IV). 4

Choose the correct answer from the options given below:

List-I	List-II
Values of $a, b$ & $c$	value of $\Delta$
(A). $a=b=c=1$	(I). 2
(B). $a=-1,b=1,c=1$	(II). 0
(C). $a=-1,b=-1,c=1$	(III). -2
(D). $a=-1,b=-1,c=-1$	(IV). 4

(A) - (IV), (B) - (I), (C) - (II), (D) - (III)

(A) - (II), (B) - (I), (C) - (IV), (D) - (III)

(A) - (II), (B) - (III), (C) - (IV), (D) - (I)

(A) - (IV), (B) - (III), (C) - (II), (D) - (I)

✅ Correct Option: 4

Expanding, $\Delta=abc+ab+bc+ca$ . For $a=b=c=1$ : $1+3=4$ (IV). For $a=-1,b=c=1$ : $-1-1=-2$ (III). For $a=b=-1,c=1$ : $1-1=0$ (II). For $a=b=c=-1$ : $-1+3=2$ (I).

2026: 19 May Shift 1

Given a matrix $A = \begin{bmatrix} 0 & 3 & -8 \\ -3 & 0 & 13 \\ 8 & -13 & 0 \end{bmatrix}$ . Which of the following statements is/are TRUE?

(A) The matrix A is a skew-symmetric matrix.

(B) The matrix A is a symmetric matrix.

(D) The matrix A is a non-singular matrix.

Choose the correct answer from the options given below:

2026: 31 May Shift 2

If $A$ and $B$ are invertible matrices of order 3. If $|A| = 2$ and $|(AB)^{-1}| = \dfrac{-1}{6}$ , then the value of $|B|$ is :

2025: 21 May Shift 2

If $x, y$ and $z$ are non-zero distinct numbers, then $\begin{vmatrix} x+y & y+z & z+x \\ z & x & y \\ 1 & 1 & 1 \end{vmatrix}$ is equal to

2025: 14 May Shift 2

The values of $\lambda$ for which the system of equation $x + 2y + z = 14, - x + y + z = 10, x + \lambda y + z = 2$ has unique solution is

2026: 22 May Shift 2

The number of diagonal matrices of order 3 with elements either 1 or 2, is:

2026: 30 May Shift 2

A is a square matrix of order 3 x 3, then

Match List-I with List-II

List-I	List-II
(A) $\left\vert \text{adj}A \right\vert$	(I) $\dfrac{1}{\left\vert A \right\vert}$
(B) $\left\vert A^{-1} \right\vert$	(II) $\left\vert A \right\vert^{4}$
(C) $\left\vert A.\text{adj}A \right\vert$	(III) $\left\vert A \right\vert^{3}$
(D) $\left\vert \text{adj}(\text{adj}A) \right\vert$	(IV) $\left\vert A \right\vert^{2}$

Choose the correct answer from the options given below:

Solution

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