JEE Main 2022 · 25 Jun · Shift 1 · Mathematics Q5

← JEE Main 2022 · 25 Jun · Shift 1

MathematicsSingle correct+4 / -1Question 5 of 90

Let A be a $3\times3$ real matrix such that
$A\begin{pmatrix}1\\1\\0\end{pmatrix}=\begin{pmatrix}1\\1\\0\end{pmatrix};\quad A\begin{pmatrix}1\\0\\1\end{pmatrix}=\begin{pmatrix}-1\\0\\1\end{pmatrix}\quad\text{and}\quad A\begin{pmatrix}0\\0\\1\end{pmatrix}=\begin{pmatrix}1\\1\\2\end{pmatrix}.$
If $X=(x_1,x_2,x_3)^T$ and I is an identity matrix of order 3, then the system $(A-2I)X=\begin{pmatrix}4\\1\\1\end{pmatrix}$ has :

no solution

infinitely many solutions

unique solution

exactly two solutions

Answer and worked solution

Official answer: B

Columns of A. $Ae_3=(1,1,2)$ ; $Ae_1=A(1,0,1)-Ae_3=(-2,-1,-1)$ ; $Ae_2=A(1,1,0)-Ae_1=(3,2,1)$ .

System. $A-2I=\begin{pmatrix}-4&3&1\\-1&0&1\\-1&1&0\end{pmatrix}$ , with determinant $4-3-1=0$ .

Consistency. The last two rows give $z=1+x$ , $y=1+x$ ; the first becomes $-4x+3+3x+1+x=4$ , true for every $x$ .

Infinitely many solutions.

(B)

Topics:Systems of linear equations Matrix algebra

← Previous question Next question →

JEE Main 2022 · 25 Jun · Shift 1, Mathematics question 5