Question

Let $I=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$ and $P=\left[\begin{array}{ccc}1& 0& 0\\ 0& -1& 0\\ 0& 0& -2\end{array}\right]$ Then the matrix $P^3+2P^2$ is equal to

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

Answer 1

The correct option is C $2I-P$

$P=\left[\begin{array}{ccc}1& 0& 0\\ 0& -1& 0\\ 0& 0& -2\end{array}\right]$

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

$=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 4\end{array}\right]$

$P^2=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 4\end{array}\right]\left[\begin{array}{ccc}1& 0& 0\\ 0& -1& 0\\ 0& 0& -2\end{array}\right]$

$=\left[\begin{array}{ccc}1& 0& 0\\ 0& -1& 0\\ 0& 0& -8\end{array}\right]$

$P^3+2P^2=\left[\begin{array}{ccc}1& 0& 0\\ 0& -1& 0\\ 0& 0& -8\end{array}\right]\left[\begin{array}{ccc}2& 0& 0\\ 0& 2& 0\\ 0& 0& 8\end{array}\right]$

$=\left[\begin{array}{ccc}3& 0& 0\\ 0& 1& 0\\ 0& 0& 0\end{array}\right]$

$P^3+2P^2\ne P,P^3+2P^2\ne 1-P$

$2I-P=\left[\begin{array}{ccc}3& 0& 0\\ 0& 1& 0\\ 0& 0& 0\end{array}\right]=P^3+2P^2$

Option C