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 p, q, r and s. Any given statement in $Column-I$ can have correct matching with $ONE$ statement in $Column-II$.

Answer 1

A) Given $|A|=162$ and A is a square matrix of order 3.
Now, $det\left\{\frac{A}{3}\right\}=\frac{1}{3^3}|A|$ (Because when we divide matrix A by 3, each element of matrix gets divided by 3. So when we take determinant of such matrix , we get $\frac{1}{3}$ common from each row (or column). So, $\frac{1}{3^3}$ multiplied by |A| )
$\Rightarrow det\left\{\frac{A}{3}\right\}=\frac{162}{27}=6$
B) Given $A^2=A$
Also $(I+A{)}^5=I+\lambda A$
${\Rightarrow }^5{C}_0{+}^5{C}_{1}A{+}^5{C}_2{A}^2{+}^5{C}_3{A}^3{+}^5{C}_4{A}^4{+}^5{C}_5{A}^5=I+\lambda A$
$\Rightarrow I+(5+10+10+5+1))A=I+\lambda A$
$\Rightarrow \lambda =31$
$\Rightarrow \frac{2\lambda +1}{7}=9$
C) Given $A=\left[\begin{array}{cc}4& 3\\ 2& 5\end{array}\right]$
$\Rightarrow A^2=\left[\begin{array}{cc}4& 3\\ 2& 5\end{array}\right]\left[\begin{array}{cc}4& 3\\ 2& 5\end{array}\right]$
$\Rightarrow A^2=\left[\begin{array}{cc}22& 27\\ 18& 31\end{array}\right]$
Also given $A^2-xA+yI=O$
$\Rightarrow \left[\begin{array}{cc}22& 27\\ 18& 31\end{array}\right]-\left[\begin{array}{cc}4x& 3x\\ 2x& 5x\end{array}\right]+\left[\begin{array}{cc}y& 0\\ 0& y\end{array}\right]=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]$
$\Rightarrow \left[\begin{array}{cc}22-4x+y& 27-3x\\ 18-2x& 31-5x+y\end{array}\right]=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]$
$\Rightarrow 22-4x+y=0;18-2x=0;27-3x=0;31-5x+y=0$
Solving these, we get
$x=9,y=14$
$\Rightarrow y-x=5$