(D) system of linear equations $AX = B$ has a unique solution if and only if the determinant of the coefficient matrix $A$ is non-zero, i.e.,$|A| \neq

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

(D) system of linear equations $AX = B$ has a unique solution if and only if the determinant of the coefficient matrix $A$ is non-zero, i.e.,$|A| \neq 0$.Given the matrix $A = \begin{bmatrix} 1 & 2 & 4 \\ 2 & 1 & 2 \\ 1 & 2 & a-4 \end{bmatrix}$.Calculating the determinant $|A|$:$|A| = 1((1)(a-4) - (2)(2)) - 2((2)(a-4) - (2)(1)) + 4((2)(2) - (1)(1))$$|A| = 1(a-4-4) - 2(2a-8-2) + 4(4-1)$$|A| = (a-8) - 2(2a-10) + 4(3)$$|A| = a - 8 - 4a + 20 + 12$$|A| = -3a + 24$For a unique solution, $|A| \neq 0$.$-3a + 24 \neq 0 \Rightarrow -3a \neq -24 \Rightarrow a \neq 8$.

Class 12 Mathematics 3 and 4 .Determinants and Matrices Solution of the Linear equations using Matrices

Easy

Under which of the following condition$(s)$ does the system of equations $\begin{bmatrix} 1 & 2 & 4 \\ 2 & 1 & 2 \\ 1 & 2 & a-4 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 6 \\ 4 \\ a \end{bmatrix}$ possess a unique solution?

WBJEE 2022 All WBJEE

A
$\forall a \in \mathbb{R}$
B
$a = 8$
C
for all integral values of $a$
D
$a \neq 8$

Explore More

Browse All

Question Bank

All Subjects

Class 12 Questions

Class 12

Mathematics Questions

Chapter

3 and 4 .Determinants and Matrices

Subtopic

Solution of the Linear equations using Matrices

Previous Year

WBJEE 2022 Paper All WBJEE years →

Let $S$ be the set of all integer solutions,$(x, y, z)$,of the system of equations
$x-2y+5z=0$
$-2x+4y+z=0$
$-7x+14y+9z=0$
such that $15 \leq x^{2}+y^{2}+z^{2} \leq 150$. Then,the number of elements in the set $S$ is equal to

DifficultJEE MAIN 2020

View Solution

If the system of linear equations $2x + 2ay + az = 0$,$2x + 3by + bz = 0$,and $2x + 4cy + cz = 0$,where $a, b, c \in R$ are non-zero and distinct,has a non-zero solution,then:

DifficultJEE MAIN 2020

View Solution

For a system of simultaneous linear equations, if $A X=\left[\begin{array}{l}1 \\ 1 \\ 2\end{array}\right]$, $\operatorname{Adj} A=\left[\begin{array}{ccc}1 & -1 & -1 \\ 1 & 1 & -1 \\ 1 & 1 & 1\end{array}\right]$ and $\operatorname{det} A>0$, then $X=$

MediumTS EAMCET 2022

View Solution

The system of equations $\begin{cases} \lambda x+y+3 z=0 \\ 2 x+\mu y-z=0 \\ 5 x+7 y+z=0 \end{cases}$ has infinitely many solutions in $\mathbb{R}$. Then,

EasyWBJEE 2019

View Solution

Consider the system of linear equations $x+y+z=4\mu$,$x+2y+2\lambda z=10\mu$,and $x+3y+4\lambda^2 z=\mu^2+15$,where $\lambda, \mu \in \mathbb{R}$. Which one of the following statements is $NOT$ correct?

DifficultJEE MAIN 2024

View Solution

Vedclass Products

For Students

Vedclass Test Series

Mock tests in real JEE/NEET style with performance analysis. 5-day free trial.

Start Free Trial

For Teachers

Exam Paper Generator

Generate Set A/B/C/D exam papers from 7.5L+ questions in 2 minutes. 3 chapters free.

Try Free

For Institutes

Online Exam Module

Live online exams with unlimited students, 360° analytics & white-label branding.

See Demo

Under which of the following condition$(s)$ does the system of equations $\begin{bmatrix} 1 & 2 & 4 \\ 2 & 1 & 2 \\ 1 & 2 & a-4 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 6 \\ 4 \\ a \end{bmatrix}$ possess a unique solution?

Similar Questions

Let $S$ be the set of all integer solutions,$(x, y, z)$,of the system of equations$x-2y+5z=0$$-2x+4y+z=0$$-7x+14y+9z=0$such that $15 \leq x^{2}+y^{2}+z^{2} \leq 150$. Then,the number of elements in the set $S$ is equal to

If the system of linear equations $2x + 2ay + az = 0$,$2x + 3by + bz = 0$,and $2x + 4cy + cz = 0$,where $a, b, c \in R$ are non-zero and distinct,has a non-zero solution,then:

For a system of simultaneous linear equations, if $A X=\left[\begin{array}{l}1 \\ 1 \\ 2\end{array}\right]$, $\operatorname{Adj} A=\left[\begin{array}{ccc}1 & -1 & -1 \\ 1 & 1 & -1 \\ 1 & 1 & 1\end{array}\right]$ and $\operatorname{det} A>0$, then $X=$

The system of equations $\begin{cases} \lambda x+y+3 z=0 \\ 2 x+\mu y-z=0 \\ 5 x+7 y+z=0 \end{cases}$ has infinitely many solutions in $\mathbb{R}$. Then,

Consider the system of linear equations $x+y+z=4\mu$,$x+2y+2\lambda z=10\mu$,and $x+3y+4\lambda^2 z=\mu^2+15$,where $\lambda, \mu \in \mathbb{R}$. Which one of the following statements is $NOT$ correct?

Vedclass Test Series

Exam Paper Generator

Online Exam Module

Let $S$ be the set of all integer solutions,$(x, y, z)$,of the system of equations
$x-2y+5z=0$
$-2x+4y+z=0$
$-7x+14y+9z=0$
such that $15 \leq x^{2}+y^{2}+z^{2} \leq 150$. Then,the number of elements in the set $S$ is equal to