The matrix \(A=\left[\begin{array}{cc}\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\ \frac{-1}{\sqrt{2}} &…
The matrix \(A=\left[\begin{array}{cc}\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\ \frac{-1}{\sqrt{2}} & \frac{-1}{\sqrt{2}}\end{array}\right]\) is
- Unitary
- Orthogonal
- Nilpotent
- Involutory
Solution
\(\begin{aligned}
A & =\left[\begin{array}{cc}
\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\
-\frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}}
\end{array}\right] \\
A^2 & =\left[\begin{array}{cc}
\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\
-\frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}}
\end{array}\right]\left[\begin{array}{cc}
\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\
-\frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}}
\end{array}\right] \\
\Rightarrow \quad A^2 & =\left[\begin{array}{cc}
\frac{1}{2}-\frac{1}{2} & \frac{1}{2}-\frac{1}{2} \\
-\frac{1}{2}+\frac{1}{2} & -\frac{1}{2}+\frac{1}{2}
\end{array}\right] \\
A^2 & =\left[\begin{array}{ll}
0 & 0 \\
0 & 0
\end{array}\right] \text { null matrix }
\end{aligned}\)
So, \(A\) is nilpotent matrix
Asked in: AP EAMCET 2020 (17 Sep Shift 1)
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