# which of the following statements is true regarding characteristics polynomials.

Let $p(x) = a_0 + a_1x + · · · + a_nx^n$ be the characteristic polynomial of a $n × n$ matrix $A$ with entries in $\mathbb{R}$. Then which of the following statements is true?
(a) $p(x)$ has no repeated roots.
(b) $p(x)$ can be expressed as a product of linear polynomials with real coefﬁcients.
(c) If $p(x)$ can be expressed as a product of linear polynomials with real coeﬃcients then there is a basis of $\mathbb{R}^n$ consisting of eigenvectors of $A$

(a)obviously $p(x)$ has repeated roots as the identity matrix has all roots as $1$.so it is false
(b)real matrix can have imaginary eigen values.so its false.
(c)Not sure.

am I right for (a) and (b). Can I get some help for (c).

## 2 Answers

A simple counter-example to (c) $$A = \left[\begin{matrix} 0 & 1 \\ 0 & 0 \end{matrix}\right]$$ The characteristic polynomial is $\lambda^{2}$, which is a product of two linear terms, but $A$ cannot be diagonalized. If it could be diagonalized, then the eigenvalues would have to be 0, which would make the diagonal matrix 0.

You are right about (a) and (b)

(c) is true if $A$ has $n$ independent eigenvectors. This is always true if the roots of $p(x)$ are distinct, i.e. not repeated. If there are repeated roots, the answer depends on the size of the Jordon blocks. If all the blocks have size 1, then the statement is true. If not, the statement is false.