- Home
- Standard 12
- Mathematics
Let $A=\left[\begin{array}{cc}i & -i \\ -i & i\end{array}\right], i=\sqrt{-1}$.Then, the system of linear equations $A^{8}\left[\begin{array}{l}x \\ y\end{array}\right]=\left[\begin{array}{c}8 \\ 64\end{array}\right]$ has :
A unique solution
Infinitely many solutions
No solution
Exactly two solutions
Solution
$A=\left[\begin{array}{cc}i & -i \\ -i & i\end{array}\right]$
$A^{2}=\left[\begin{array}{cc}-2 & 2 \\ 2 & -2\end{array}\right]=2\left[\begin{array}{cc}-1 & 1 \\ 1 & -1\end{array}\right]$
$A^{4}=2^{2}\left[\begin{array}{cc}2 & -2 \\ -2 & 2\end{array}\right]=8\left[\begin{array}{cc}1 & -1 \\ -1 & 1\end{array}\right]$
$A^{8}=64\left[\begin{array}{cc}2 & -2 \\ -2 & 2\end{array}\right]=128\left[\begin{array}{cc}1 & -1 \\ -1 & 1\end{array}\right]$
$A^{8}\left[\begin{array}{l}x \\ y\end{array}\right]=\left[\begin{array}{l}8 \\ 64\end{array}\right]$
$\Rightarrow 128\left[\begin{array}{cc}1 & -1 \\ -1 & 1\end{array}\right]\left[\begin{array}{l} x \\ y \end{array}\right]=\left[\begin{array}{c}8 \\ 64\end{array}\right]$
$\Rightarrow \quad 128\left[\begin{array}{c}x-y \\ -x+y\end{array}\right]=\left[\begin{array}{c}8 \\ 64\end{array}\right]$
$\Rightarrow \quad x-y=\frac{1}{16}………(1)$
$\quad-x+y=\frac{1}{2}$ $…..(2)$
$\Rightarrow$ From $(1)$ and $(2)$ No solution.
Similar Questions
Consider the following information regarding the number of men and women workers in three factories $I,\,II$ and $III$
| Men workers |
Women workers |
|
| $I$ | $30$ | $25$ |
| $II$ | $25$ | $31$ |
| $III$ | $27$ | $26$ |
Represent the above information in the form of a $3 \times 2$ matrix. What does the entry in the third row and second column represent?
