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## Homework Statement

A is a given 3x3 matrix

A=[tex]

\begin{pmatrix}

a & 0 & 1 \\

0 & a & b \\

0 & 0 & c

\end{pmatrix}

[/tex]

a,b,c are real numbers

1) determine for what values of a,b,c the matrix A is diagnoizable. (advice: distinguish between the cases a=c and a≠c)

## Homework Equations

characteristic polynomial of A is det(tI-A)

## The Attempt at a Solution

finding the solution for det(tI-A)=0

tI-A=[tex]

\begin{pmatrix}

t-a & 0 & -1 \\

0 & t-a & -b \\

0 & 0 & t-c

\end{pmatrix}

[/tex]

the characteristic polynomial is (t-c)(t-a)^2 and therefore the eigenvalues are a and c.

for t=c we get the matrix [tex]

\begin{pmatrix}

c-a & 0 & -1 \\

0 & c-a & -b \\

0 & 0 & 0

\end{pmatrix}

[/tex]

let it be C,

__x__=(x,y,z), for C

__x__=

__0__we get the system of linear equations:

(c-a)x-z=0

(c-a)y -bz=0

likewise, for t=a we get

-1z=0

-bz=0

(a-c)z=0

after that I'm pretty much lost. my guess is that you need to find the dimension of the eigenspace that corresponds to each eigenvalue, but I'm nut sure how.

Help would be very welcomed