Answer to Question #254843 in Linear Algebra for Sabelo Xulu

Question

Suppose A is a matrix with characteristic polynomial p($\lambda$ ) =$\lambda$^3 - $\lambda$

a) What is the order of the matrix A?

b) Is A invertible?

c) Is A diagonalisable?

d) Find the eigenvalues of A²

Answer 1

a) The characteristics

polynomial P($\lambda$ ) has degree n

$\therefore$ A is of order $3×3$

b)The roots of characteristic polynomial

$\lambda$ ³ -$\lambda=0$ are

$\lambda(\lambda$ ²$-1)=0$

$\implies \lambda(\lambda-1)(\lambda+1)=0$

$\therefore \lambda=0,$ or $-1$ or $1$

A matrix is invertible $\iff$

$det(A) \mathrlap{\,/}{=}$

det A is exactly the product of A eigenvalues

$0.-1.1=0$

Since the product is

Then $A$ is not invertible

c) A is diagonalisable because the characteristic polynomial can be factored into distinct linear factors

I.e $\lambda$ ³ - $\lambda=\lambda(\lambda$ ² -$1$ )

=$\lambda (\lambda-1)(\lambda+1)$

d) Eigenvalues of $A$ ² are $(0)$ ²,$(-1)$ ² and $(1)$ ²

Which are

$0,1$ and $1$