Question

If $A$ is a square matrix of order $3,$ then the degree of characteristics polynomial for matrix $A,$ will be

• A. $0$ always
• B. $1$ always
• C. $2$ always
• D. $3$ always.

Answer 1

The correct option is D $3$ always.

Let $A=\left[\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right],$ then characteristics polynomial is
$f\left(\lambda \right)=|A-\lambda I|=\left|\begin{array}{ccc}a-\lambda & b& c\\ d& e-\lambda & f\\ g& h& i-\lambda \end{array}\right|=(a-\lambda )[{\lambda }^2-(i+e\left)\lambda \right)+ie-hf]-b[d(i-\lambda )-gf]+c[dh-eg+g\lambda ]$
Clearly, the degree of characteristics polynomial for matrix $A$ is $3$.