Let A be a 3 3 real matrix such that A ( l 1 1 0 )= ( l 1 1 0 ) ;… - Vidyadip

MCQ

Let $A$ be a $3 \times 3$ real matrix such that $A \left(\begin{array}{l}1 \\ 1 \\ 0\end{array}\right)=\left(\begin{array}{l}1 \\ 1 \\ 0\end{array}\right) ; A \left(\begin{array}{l}1 \\ 0 \\ 1\end{array}\right)=\left(\begin{array}{c}-1 \\ 0 \\ 1\end{array}\right)$ and $A \left(\begin{array}{l}0 \\ 0 \\ 1\end{array}\right)=\left(\begin{array}{l}1 \\ 1 \\ 2\end{array}\right)$. If $X =\left( x _{1}, x _{2}, x _{3}\right)^{ T }$ and $I$ is an identity matrix of order $3$ , then the system $( A -2 I ) X =\left(\begin{array}{l}4 \\ 1 \\ 1\end{array}\right)$ has

A
no solution
✓
infinitely many solutions
C
unique solution
D
exactly two solutions

✓

Answer

Correct option: B.

infinitely many solutions

b
$A =\left[\begin{array}{lll} a _{1} & b _{1} & c _{1} \\ a _{2} & b _{2} & c _{2} \\ a _{3} & b _{3} & c _{3}\end{array}\right]$

$A \left[\begin{array}{l}0 \\ 0 \\ 1\end{array}\right]=\left[\begin{array}{l} c _{1} \\ c _{2} \\ c _{3}\end{array}\right]=\left[\begin{array}{l}1 \\ 1 \\ 2\end{array}\right]$

$\Rightarrow c _{1}=1, c _{2}=1, c _{3}=2$

$A \left[\begin{array}{l}1 \\ 0 \\ 1\end{array}\right]=\left[\begin{array}{l} c _{1}+ a _{1} \\ c _{2}+ a _{2} \\ c _{3}+ a _{3}\end{array}\right]=\left[\begin{array}{c}-1 \\ 0 \\ 1\end{array}\right]$

$\Rightarrow a _{1}=-2, a _{2}=-1, a _{3}=-1$

$A \left[\begin{array}{l}1 \\ 1 \\ 0\end{array}\right]=\left[\begin{array}{l} a _{1}+ b _{1} \\ a _{2}+ b _{2} \\ a _{3}+ b _{3}\end{array}\right]=\left[\begin{array}{l}1 \\ 1 \\ 0\end{array}\right]$

$\Rightarrow b _{1}=3, b _{2}=2, b _{3}=1$

$\Rightarrow \quad A =\left[\begin{array}{lll}-2 & 3 & 1 \\ -1 & 2 & 1 \\ -1 & 1 & 2\end{array}\right]$

$\Rightarrow A -2 I =\left[\begin{array}{ccc}-4 & 3 & 1 \\ -1 & 0 & 1 \\ -1 & 1 & 0\end{array}\right]$

$| A -2 I |=0$

Now, $\left[\begin{array}{lll}-4 & 3 & 1 \\ -1 & 0 & 1 \\ -1 & 1 & 0\end{array}\right]\left[\begin{array}{l} x _{1} \\ x _{2} \\ x _{3}\end{array}\right]=\left[\begin{array}{l}4 \\ 1 \\ 1\end{array}\right]$

$-4 x_{1}+3 x_{2}+x_{3}=4 \quad \ldots .$ ($1$)

$- x _{1}+ x _{3}=1 \ldots .$ ($2$)

$- x _{1}+ x _{2}=1 \ldots .$ ($2$)

(1) $-[(2)+3(3)]$

$0=0 \Rightarrow$ infinite solutions

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "M.C.Q (1 Marks)" from 3 and 4 . determinant and metrices→3 and 4 . determinant and metrices — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Vectors $a, b, c$ are inclined to each other at an angle of ${60^o}$ and $|a|\, = \,|b|\, = 2$ and $|c|\, = 2,$ then $(2a + 3b - 5c)\,.\,\,(4a - 6b + 10c) = $

$\lim _{x \rightarrow \frac{\pi}{2}}\left(\frac{1}{\left(x-\frac{\pi}{2}\right)^2} \int_{x^3}^{\left(\frac{\pi}{2}\right)^3} \cos \left(\frac{1}{t^3}\right) d t\right)$ is equal to

If $f(x) = x^2 + 2bx + 2c^2$ and $g(x) = -x^2 -2cx + b^2$ such that $\min . f(x) > \max . g(x),$ then relation between $b$ and $c$ is

If $\frac{{d[f(x)]}}{{dx}} = g(x)$ for $a \le x \le b,$ then $\int_a^b {f(x)\,\,g(x)\,dx} $ equals

The value of $\int\limits_{0}^{2 \pi} \frac{x \sin ^{8} x}{\sin ^{8} x+\cos ^{8} x} d x$ is equal to

The function $f(x) = \frac{{{{\sec }^{ - 1}}x}}{{\sqrt {x - [x]} }},$ where $[.]$ denotes the greatest integer less than or equal to $x$ is defined for all $x$ belonging to

$\int_{}^{} {\frac{{{x^4} + {x^2} + 1}}{{{x^2} - x + 1}}\;dx = } $

${d \over {dx}}\left[ {\log \left\{ {{e^x}{{\left( {{{x - 2} \over {x + 2}}} \right)}^{3/4}}} \right\}} \right]$ is equals to

If $\text{x}=\sin ^{ -1 }{ \text{K} },\text{y}=\cos ^{ -1 }\text{K}, -1\le \text{K}\le 1$, then the correct relationship is:

$\text{x}+\text{y}=\frac{\pi}{8}$
$\text{x}+\text{y}={2}$
$\text{x}+\text{y}=\frac{\pi}{2}$
$\text{x}+\text{y}=\frac{\pi}{8}$

Given a system of inequation: $2\text{y}-\text{x}\leq4$ $-2\text{x}+\text{y}\geq-4$Find the value of s, which is the greatest possible sum of the x and y co - ordinates of the point which satisfies the following inequalities as graphed in the xy plane.