Question

Following questions contains statements given in two columns, which have to be matched. The statements in $Column-I$ are labelled as A, B, C and D while the statements in $Column-II$ are labelled as 1,2,3 and 4. Any given statement in $Column-I$ can have correct matching with $ONE$ statement in $Column-II$.

Answer 1

A) Let $A=\left[\begin{array}{ccc}2& -2& -4\\ -1& 3& 4\\ 1& -2& -3\end{array}\right]$
Now, $A^2=\left[\begin{array}{ccc}2& -2& -4\\ -1& 3& 4\\ 1& -2& -3\end{array}\right]\left[\begin{array}{ccc}2& -2& -4\\ -1& 3& 4\\ 1& -2& -3\end{array}\right]$
$\Rightarrow A^2=\left[\begin{array}{ccc}2& -2& -4\\ -1& 3& 4\\ 1& -2& -3\end{array}\right]$
$\Rightarrow A^2=A$
Hence,A is an idempotent matrix.
B) Let $A=\left[\begin{array}{ccc}-5& -8& 0\\ 3& 5& 0\\ 1& 2& -1\end{array}\right]$
$A^2=\left[\begin{array}{ccc}-5& -8& 0\\ 3& 5& 0\\ 1& 2& -1\end{array}\right]\left[\begin{array}{ccc}-5& -8& 0\\ 3& 5& 0\\ 1& 2& -1\end{array}\right]$
$\Rightarrow A^2=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$
$\Rightarrow A^2=I$
Hence, A is involutary matrix.
C) Let $A=\frac{1}{3}\left[\begin{array}{ccc}1& -2& 2\\ -2& 1& 2\\ -2& -2& -1\end{array}\right]$
$\Rightarrow A^T=\frac{1}{3}\left[\begin{array}{ccc}1& -2& -2\\ -2& 1& -2\\ 2& 2& -1\end{array}\right]$
Now, $A^{T}A=\frac{1}{9}\left[\begin{array}{ccc}1& -2& -2\\ -2& 1& -2\\ 2& 2& -1\end{array}\right]\left[\begin{array}{ccc}1& -2& 2\\ -2& 1& 2\\ -2& -2& -1\end{array}\right]$
$=\frac{1}{9}\left[\begin{array}{ccc}9& 0& 0\\ 0& 9& 0\\ 0& 0& 9\end{array}\right]$
$\Rightarrow A^{T}A=I$
Hence, A is orthogonal matrix
D) Let $A=\left[\begin{array}{ccc}1& 1& 3\\ 5& 2& 6\\ -2& -1& -3\end{array}\right]$
$A^2=\left[\begin{array}{ccc}1& 1& 3\\ 5& 2& 6\\ -2& -1& -3\end{array}\right]\left[\begin{array}{ccc}1& 1& 3\\ 5& 2& 6\\ -2& -1& -3\end{array}\right]$
$\Rightarrow A^2=\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]$
$\Rightarrow A^2=O$ where O is zero matrix.
Hence, A is a nilpotent matrix of order 2.