Question

If $I=\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)P=\left(\begin{array}{ccc}1& 0& 0\\ 0& -1& 0\\ 0& 0& -1\end{array}\right)$. Then the matrix $P^3+2P^2$ is equal to

• A. $P$
• B. $I-P$
• C. $2I+P$
• D. $2I-P$

Answer 1

The correct option is C $2I+P$

Given,

$P=\left(\begin{array}{ccc}1& 0& 0\\ 0& -1& 0\\ 0& 0& -1\end{array}\right)$.

Now

$P^2=\left(\begin{array}{ccc}1& 0& 0\\ 0& -1& 0\\ 0& 0& -1\end{array}\right)$ $\left(\begin{array}{ccc}1& 0& 0\\ 0& -1& 0\\ 0& 0& -1\end{array}\right)$

or, $P^2=$ $\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)=I$......(1).

Now, $P^3=P$ [ From (1) we get]......(2).

Now $P^3+2P^2$

$=P+2I$ [Using (1) and (2)]