Let A = [ cases i & - i - i & i cases ], i = - 1 . Then, the system of linear equations A ^ 8 [ array ll x y array ] = [ array ll 8 64 array ] has :

Correct answer: C. No solution A = [ cases i & - i - i & i cases ] A ^ 2 = [ cases - 2 & 2 2 & - 2 cases ] = 2[ cases - 1 & 1 1 & - 1 cases ] A ^ 4 = 2 ^ 2 [ cases 2 & - 2 - 2 & 2 cases ] = 8[ cases 1 & - 1 - 1 & 1 cases ] A ^ 8 = 64[ cases 2 & - 2 - 2 & 2 cases ] = 128[ cases 1 & - 1 - 1 & 1 cases ] A ^ 8 [ array ll x y array ] = [ array ll 8 64 array ] ⇒ 128[ cases 1 & - 1 - 1 & 1 cases ][ array ll x y array ] = [ array ll 8 64 array ] ⇒ 128[ array ll x - y - x + y array ] = [ array ll 8 64 array ] ⇒ x - y = (1)/(16) . . . ( 1 ) and - x + y = (1)/(2) . . . ( 2 ) ⇒ From ( 1 ) and ( 2 ) : No solution.

JEE Main 202116 MarchMorning Shift

Exam: JEE Main
Year: 2021
Date: 16 March
Shift: Morning Shift
Subject: Mathematics
Chapter: Matrices

JEE Main 2021 · 16 March · Morning Shift · Mathematics · Matrices

Let A = [ cases i & - i - i & i cases ], i = - 1 . Then, the system...

Let

A = \left[

\begin{cases}i & - i \\ - i & i\end{cases}

\right], i = \sqrt{- 1}

. Then, the system of linear equations

{A}^{8}\left[

\begin{array}{ll}x \\ y\end{array}

\right] = \left[

\begin{array}{ll}8 \\ 64\end{array}

\right]

has :

Options

AA unique solution
BInfinitely many solutions
CNo solution
DExactly two solutions

Correct answer

C. No solution

Step-by-step solution

$A = \left[$ \begin{cases}i & - i \\ - i & i\end{cases} $\right]$

${A}^{2} = \left[$ \begin{cases}- 2 & 2 \\ 2 & - 2\end{cases} $\right] = 2\left[$ \begin{cases}- 1 & 1 \\ 1 & - 1\end{cases} $\right]$

${A}^{4} = {2}^{2}\left[$ \begin{cases}2 & - 2 \\ - 2 & 2\end{cases} $\right] = 8\left[$ \begin{cases}1 & - 1 \\ - 1 & 1\end{cases} $\right]$

${A}^{8} = 64\left[$ \begin{cases}2 & - 2 \\ - 2 & 2\end{cases} $\right] = 128\left[$ \begin{cases}1 & - 1 \\ - 1 & 1\end{cases} $\right]$

${A}^{8}\left[$ $\begin{array}{ll}x \\ y\end{array}$ $\right] = \left[$ $\begin{array}{ll}8 \\ 64\end{array}$ $\right]$

$⇒ 128\left[$ \begin{cases}1 & - 1 \\ - 1 & 1\end{cases} $\right]\left[$ $\begin{array}{ll}x \\ y\end{array}$ $\right] = \left[$ $\begin{array}{ll}8 \\ 64\end{array}$ $\right]$

$⇒ 128\left[$ $\begin{array}{ll}x - y \\ - x + y\end{array}$ $\right] = \left[$ $\begin{array}{ll}8 \\ 64\end{array}$ $\right]$

$⇒ x - y = \dfrac{1}{16}$ $. . . ( 1 )$

and $- x + y = \dfrac{1}{2}$ $. . . ( 2 )$

$⇒$ From $( 1 )$ and $( 2 )$ : No solution.

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