Question

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

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

$P^3=P\times P\times P$

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

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

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

$P^3+2P^2=\left[\begin{array}{ccc}1& 0& 0\\ 0& -1& 0\\ 0& 0& -4\end{array}\right]+\left[\begin{array}{ccc}2& 0& 0\\ 0& 2& 0\\ 0& 0& 4\end{array}\right]$

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

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

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

$=2I+P$