Question

The characteristic equation of a $3\times 3$ matrix P is defined as
$a\left(\lambda \right)=|\lambda I-P|={\lambda }^3+{\lambda }^2+2\lambda +1=0$
If I denotes identify matrix, then the inverse of matrix P will be

• A. $(P^2+P+2I)$
• B. $(P^2+P+1)$
• C. $-(P^2+P+I)$
• D. $-(P^2+P+2I)$

Answer 1

The correct option is D $-(P^2+P+2I)$

According to Cayley-Hamilton theorem, every square matrix satisfies its own characteristic equation
$\Rightarrow P^3+P^2+2P+1=O$
Multiplying by matrix $[P{]}^{-1}$
$\Rightarrow P^2+P+2I+P^{-1}=0$
$\Rightarrow P^{-1}=-(P^2+P+2I)$