If A, B and C are three singular matrices given by A=[arrayll1 & 4 \\ 3 & 2 aarray], B=[arrayll3 b & 5 \\ a & 2array] and C=[arraycca+b+c & c+1 \\ a+c & carray], then the value of a b c is : Option 1: 15 Option 2: 30 Option 3: 45 Option 4: 90

Answer: 45. Solution: To find the value of abc, we need to find the values of a, b, and c using the fact that all three matrices are singular. A matrix is singular when its determinant equals zero. For matrix A = bmatrix 1 & 4 \\ 3 & 2a bmatrix: (A) = 1 x 2a - 4 x 3 = 0 2a - 12 = 0 a = 6 For matrix B = bmatrix 3b & 5 \\ a & 2 bmatrix, using a = 6: (B) = 3b x 2 - 5 x 6 = 0 6b - 30 = 0 b = 5 For matrix C = bmatrix a+b+c & c+1 \\ a+c & c bmatrix, substituting a = 6 and b = 5: (C) = (11+c)c - (c+1)(6+c) = 0 11c + c^2 - 6c - 6 - c^2 - c = 0 4c - 6 = 0 c = 6/4 = 3/2 Therefore: abc = 6 x 5 x 3/2 = 30 x 3/2 = 45

Algebra

Matrices & Determinants

Medium

core

If $A, B$ and $C$ are three singular matrices given by $A=\left[\begin{array}{ll}1 & 4 \\ 3 & 2 a\end{array}\right], B=\left[\begin{array}{ll}3 b & 5 \\ a & 2\end{array}\right]$ and $C=\left[\begin{array}{cc}a+b+c & c+1 \\ a+c & c\end{array}\right]$ , then the value of $a b c$ is :

✅ Correct Option: 3

To find the value of $abc$ , we need to find the values of $a$ , $b$ , and $c$ using the fact that all three matrices are singular.

A matrix is singular when its determinant equals zero.

For matrix $A = \begin{bmatrix} 1 & 4 \\ 3 & 2a \end{bmatrix}$ :

$\det(A) = 1 \times 2a - 4 \times 3 = 0$

$2a - 12 = 0$

$a = 6$

For matrix $B = \begin{bmatrix} 3b & 5 \\ a & 2 \end{bmatrix}$ , using $a = 6$ :

$\det(B) = 3b \times 2 - 5 \times 6 = 0$

$6b - 30 = 0$

$b = 5$

For matrix $C = \begin{bmatrix} a+b+c & c+1 \\ a+c & c \end{bmatrix}$ , substituting $a = 6$ and $b = 5$ :

$\det(C) = (11+c)c - (c+1)(6+c) = 0$

$11c + c^2 - 6c - 6 - c^2 - c = 0$

$4c - 6 = 0$

$c = \dfrac{6}{4} = \dfrac{3}{2}$

Therefore:

$abc = 6 \times 5 \times \dfrac{3}{2} = 30 \times \dfrac{3}{2} = 45$

2026: 21 May Shift 2

The value of $\begin{vmatrix}2 & 3 & 4\\5 & 6 & 8\\6x & 9x & 12x\end{vmatrix}$ is

2026: 25 May Shift 1

The value of $\Delta = \begin{vmatrix} 2\sin40^\circ & 2\sin50^\circ \\ -3\cos40^\circ & 3\cos50^\circ \end{vmatrix}$ is:

2026: 22 May Shift 1

If A is a square matrix such that $A^2=A$ and $I$ is identity matrix of same order, then $(2I+A)^3-19A$ is equal to:

2026: 29 May Shift 2

If $\begin{vmatrix}x&2\\18&x\end{vmatrix} = \begin{vmatrix}6&2\\18&6\end{vmatrix}$ , then x equals:

2026: 22 May Shift 2

If $A=\begin{bmatrix}k & 2\\2 & k\end{bmatrix}$ such that $\lvert A^3 \rvert = 125$ then which of the following statements are TRUE?

(A) $\lvert k.adj(3A) \rvert = 405$

(B) $\lvert k.adj(3A) \rvert = 2025$

(D) $\lvert adj(3A) \rvert = 45$

Choose the correct answer from the options given below:

2026: 31 May Shift 2

If $\begin{vmatrix} 1 & -2 & 5 \\ 2 & \lambda & -1 \\ 0 & 4 & 2\lambda \end{vmatrix} = 86$ , the sum of the values of $\lambda$ is:

Solution

Related questions:

Keyboard Shortcuts

CUET Mathematics Past Year Questions (Topic-Wise)