Find missing Eigenvalues and check invertability

Question

I've been struggling with this problem for a couple of days now. I'm familiar with matrices and how to find eigenvalues and eigenvectors to write them as $QDQ^{-1}$. But I don't seem to able to crack this one. Can somebody maybe help out?

Given:

$\lambda_1+\lambda_2+\lambda_3=\frac{2}{3}$

$a_{1,1}+a_{2,2}+a_{3,3}=\frac{2}{3}$

$A * \begin{pmatrix}p\\q\\r\\\end{pmatrix} = 0$

$\lambda_1 =$ ? with eigenvector $v_1 = (p, q, r)$

$\lambda_2 = 1$ with eigenvector $v_2 = (1,0,0)$

$\lambda_3 =$ ? with eigenvector $v_3 = (-15, 4, 0)$

Find:

• Find the missing eigenvalues $\lambda_1$ and $\lambda_2$.
• Is matrix A diagonalizable?
• Is matrix A invertable?

Answer 1

As @mathlover said: because of $Av = \lambda v \to \lambda_1=0$

Since $\lambda_1+\lambda_2+\lambda_3=\frac{2}{3} \to \lambda_3 = -\frac{1}{3}$

An n by n matrix A is diagonalisable if and only if a A has n lineair undependent eigenvectors.

$\to$ A is diagonalisable.

A n by n matrix A is invertable if and only if 0 is not a Eigenvector of A

$\to$ Since $\lambda_1 = 0$, A is not invertable.