For the system of linear equations a x+y+z=1, | vedclass

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

Question

(A) The determinant of the coefficient matrix is $D = \begin{vmatrix} a & 1 & 1 \ 1 & a & 1 \ 1 & 1 & a \end{vmatrix} = a(a^2-1) - 1(a-1) + 1(1-a) =

Vedclass · Accepted Answer

It has infinitely many solutions if $a=2$ and $\beta=-1$

Vedclass · Answer

(A) The determinant of the coefficient matrix is $D = \begin{vmatrix} a & 1 & 1 \ 1 & a & 1 \ 1 & 1 & a \end{vmatrix} = a(a^2-1) - 1(a-1) + 1(1-a) = a^3 - 3a + 2 = (a-1)^2(a+2)$.For $a=1$,the equations become $x+y+z=1$,$x+y+z=1$,$x+y+z=\beta$. If $\beta=1$,there are infinitely many solutions. Thus,option $D$ is correct.For $a=-2$,$D=0$. The augmented matrix for $\beta=1$ is $\begin{bmatrix} -2 & 1 & 1 & 1 \ 1 & -2 & 1 & 1 \ 1 & 1 & -2 & 1 \end{bmatrix}$. Adding the rows gives $0=3$,which is impossible. Thus,there is no solution. Option $B$ is correct.For $a=2$ and $\beta=1$,$D = (2-1)^2(2+2) = 4 
eq 0$. The system has a unique solution. Using Cramer's rule,$x=y=z = \frac{1}{4}$. Thus $x+y+z = \frac{3}{4}$. Option $C$ is correct.For $a=2$ and $\beta=-1$,$D=4 
eq 0$,so the system has a unique solution,not infinitely many. Thus,option $A$ is incorrect.

For the system of linear equations $a x+y+z=1$,$x+a y+z=1$,$x+y+a z=\beta$,which one of the following statements is $NOT$ correct?

Similar Questions

Let $X = \begin{bmatrix} a \\ b \\ c \end{bmatrix}$,$A = \begin{bmatrix} 1 & -1 & 2 \\ 2 & 0 & 1 \\ 3 & 2 & 1 \end{bmatrix}$,and $B = \begin{bmatrix} 3 \\ 1 \\ 4 \end{bmatrix}$. If $AX = B$,then the value of $2a - 3b + 4c$ is:

If the system of equations
$x-2y+3z=9$
$2x+y+z=b$
$x-7y+az=24$
has infinitely many solutions,then $a-b$ 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:

The bookshop of a particular school has $10$ dozen chemistry books,$8$ dozen physics books,and $10$ dozen economics books. Their selling prices are Rs. $80$,Rs. $60$,and Rs. $40$ each respectively. Find the total amount the bookshop will receive from selling all the books using matrix algebra.

Vedclass Test Series

Exam Paper Generator

Online Exam Module