1. ## Theory of Matrices

Let A belonging $\displaystyle M_n$ be a non-singular matrix. Let $\displaystyle p_A(t)$ be its characteristic polynomial, $\displaystyle p_A(t)$ = det(tI-A) = $\displaystyle t^n + a_(n-1)t^(n-1) + ... + a_1t + a_0$.

a) Prove that $\displaystyle a_0 = (-1)^n det(A)$

2. ## Re: Theory of Matrices

Originally Posted by page929
Let A belonging $\displaystyle M_n$ be a non-singular matrix. Let $\displaystyle p_A(t)$ be its characteristic polynomial, $\displaystyle p_A(t)$ = det(tI-A) = $\displaystyle t^n + a_(n-1)t^(n-1) + ... + a_1t + a_0$.

a) Prove that $\displaystyle a_0 = (-1)^n det(A)$

$\displaystyle a_0=p_A(0)=\det (0I-A)=\det(-A)$ . Now, note that $\displaystyle -A$ is the result of changing the sign to all columns of A.

P.S. The hypothesis $\displaystyle A$ non-singular is irrelevant.

3. ## Re: Theory of Matrices

By the way, to get all of a subscript or all of a superscript above or below the main level, in LaTeX, put them in { }: e^{ax_{1+1}} gives $\displaystyle e^{ax_{1+1}}$.

4. ## Re: Theory of Matrices

Originally Posted by FernandoRevilla
$\displaystyle a_0=p_A(0)=\det (0I-A)=\det(-A)$ . Now, note that $\displaystyle -A$ is the result of changing the sign to all columns of A.
or, det(-A) = det[(-I)(A)] = det(-I)xdet(A) = ((-1)^n)xdet(A)