R S Aggarwal Solutions for Exercise 3: EXERCISE

If $A=\left[\begin{array}{ll}3& -2\\ 4& -2\end{array}\right]$, find $k$ so that $A^2=kA-2I$

If $\left[\begin{array}{lll}1& x& 1\end{array}\right]\left[\begin{array}{lll}1& 2& 3\\ 4& 5& 6\\ 3& 2& 5\end{array}\right]\left[\begin{array}{c}1\\ -2\\ 3\end{array}\right]=0$, find $x$.

If $A=\left[\begin{array}{cc}3& 4\\ -4& -3\end{array}\right]$, find $f\left(A\right)$, where $f\left(x\right)=x^2–5x+7$.

If $A=\left[\begin{array}{ll}1& 1\\ 0& 1\end{array}\right]$, prove that $A^n=\left[\begin{array}{ll}1& n\\ 0& 1\end{array}\right]$ for all $n\in N.$

Given an example of two matrices $A$ and $B$ such that $A\ne 0, B\ne 0, AB=0 \mathrm{and} BA\ne 0$.

Give an example of three matrices $A, B ,C$ such that $AB=AC$ but $B\ne C$.

If $A=\left[\begin{array}{cc}1& 0\\ -1& 7\end{array}\right]\text{ and }B=\left[\begin{array}{cc}0& 4\\ -1& 7\end{array}\right]$, find $\left(3A^2-2B+1\right)$.

If $\left[\begin{array}{cc}2& 3\\ 5& 7\end{array}\right]\left[\begin{array}{cc}1& -3\\ -2& 4\end{array}\right]=\left[\begin{array}{cc}-4& 6\\ -9& x\end{array}\right]$, find the value of $x$.