Suppose the vectors $x_1$, $x_2$ and $x_3$ are the solutions of the system of linear equations, $Ax=b$ when the vector $b$ on the right side is equal to $b_1$, $b_2$ and $b_3$ respectively. If $x_1=\begin{bmatrix}1\\1\\1\end{bmatrix}$, $x_2=\begin{bmatrix}0\\2\\1\end{bmatrix}$, $x_3=\begin{bmatrix}0\\0\\1\end{bmatrix}$, $b_1=\begin{bmatrix}1\\0\\0\end{bmatrix}$, $b_2=\begin{bmatrix}0\\2\\0\end{bmatrix}$ and $b_3=\begin{bmatrix}0\\0\\2\end{bmatrix}$, then the determinant of $A$ is equal to:

JEE Main · 2020 · 04 Sep 2020 · Shift 2 · Matrices

Suppose the vectors

x_{1}

x_{2}

and

x_{3}

are the solutions of the system of linear equations,

A x = b

when the vector

b

on the right side is equal to

b_{1}

b_{2}

and

b_{3}

respectively. If

x_{1} = 111

x_{2} = 021

x_{3} = 001

b_{1} = 100

b_{2} = 020

and

b_{3} = 002

, then the determinant of

A

is equal to:

Answer: (C)

Since

A x_{i} = b_{i}

in the three cases, collecting the solution vectors as the columns of

X = [x_{1} x_{2} x_{3}]

and the right sides as the columns of

B = [b_{1} b_{2} b_{3}]

gives

A X = B

, so

det A = \frac{det B}{det X}

. Here

det X = 111021001 = 2

and

det B = 100020002 = 4

, so

det A = \frac{4}{2} = 2

JEE Main · 2020 · 04 Sep 2020 · Shift 2 · Single correct option