If the rank of the matrix begin{bmatrix} x & | vedclass

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

Question

(C) Let $A = \begin{bmatrix} x & x & x \ x & x^2 & x \ x & x & x+1 \end{bmatrix}$.For the rank of $A$ to be $1$, all minors of order $2$ must be zer

Vedclass · Accepted Answer

$x=0$

Vedclass · Answer

(C) Let $A = \begin{bmatrix} x & x & x \ x & x^2 & x \ x & x & x+1 \end{bmatrix}$.For the rank of $A$ to be $1$, all minors of order $2$ must be zero.Consider the minor formed by the first two rows and first two columns: $\begin{vmatrix} x & x \ x & x^2 \end{vmatrix} = x^3 - x^2 = x^2(x-1)$.For this to be $0$, $x=0$ or $x=1$.Case $1$: If $x=1$, $A = \begin{bmatrix} 1 & 1 & 1 \ 1 & 1 & 1 \ 1 & 1 & 2 \end{bmatrix}$. The minor $\begin{vmatrix} 1 & 1 \ 1 & 2 \end{vmatrix} = 2-1 = 1 
eq 0$. Thus, the rank is at least $2$. So $x=1$ is not the solution.Case $2$: If $x=0$, $A = \begin{bmatrix} 0 & 0 & 0 \ 0 & 0 & 0 \ 0 & 0 & 1 \end{bmatrix}$. All minors of order $2$ are zero, and there is at least one non-zero element (the $1$ at $A_{33}$). Thus, the rank is $1$.Therefore, the only possible value is $x=0$.

If the rank of the matrix $\begin{bmatrix} x & x & x \\ x & x^2 & x \\ x & x & x+1 \end{bmatrix}$ is $1$, then:

Similar Questions

The rank of the matrix $\begin{bmatrix} 4 & 2 & 1-x \\ 5 & k & 1 \\ 6 & 3 & 1+x \end{bmatrix}$ is $1$,then

If ${D_p} = \begin{vmatrix} p & 15 & 8 \\ p^2 & 35 & 9 \\ p^3 & 25 & 10 \end{vmatrix}$,then ${D_1} + {D_2} + {D_3} + {D_4} + {D_5} = $

If $f(x) = \begin{vmatrix} x^3-x & 2e^{2x} & \sin x^2 \\ \cos(2x) & x+x^2 & e^{-x} \\ \tan 3x & \ln(1-2x) & x^2+x+1 \end{vmatrix}$,then $f'(0)$ is equal to:

If $D(x) = \begin{vmatrix} x - 1 & (x - 1)^2 & x^3 \\ x - 1 & x^2 & (x + 1)^3 \\ x & (x + 1)^2 & (x + 1)^3 \end{vmatrix}$,then the coefficient of $x$ in $D(x)$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module