Q93 / 200 · WBJEE 2013 · Mathematics

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

  • A
    P
  • B
    I-P
  • Correct answer
    C
    $2I+P$
  • D
    $2I-P$

Solution

Given, $I=\left(\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right)$ and $P=\left(\begin{array}{ccc}1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -2\end{array}\right)$ The characteristic equation of $P$ is $$ |P-\lambda|=0 $$ $\Rightarrow\left|\begin{array}{ccc}1-\lambda & 0 & 0 \\ 0 & -1-\lambda & 0 \\ 0 & 0 & -2-\lambda\end{array}\right|=0$ $\Rightarrow \quad(1-\lambda)\{(1+\lambda)(2+\lambda)\}=0$ $\Rightarrow \quad\left(1-\lambda^{2}\right)(2+\lambda)=0$ $\Rightarrow \quad 2-2 \lambda^{2}+\lambda-\lambda^{3}=0$ $\Rightarrow \quad \lambda^{3}+2 \lambda^{2}-\lambda-2=0$ We know that, Caylay Hamilton theorem states that Every square matrix satisfy its characteristic equation'. $\therefore$ $$ P^{3}+2 P^{2}-P-2 I=0 $$ $\Rightarrow$\(p^{3}+2 p^{2}=2 I+p\)

Hence, the correct answer is option: 'C'.

Relevant Questions