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## Homework Statement

For the following matrix, find the value of t, if any, so that the following matrix is diagonalizable

[tex]A=

\begin{pmatrix}

5 & -2 & 4\\

0 & 3 & t\\

0 & 0 & 5

\end{pmatrix}

[/tex]

## The Attempt at a Solution

In order for A to be diagonalizable, we need 3 linearly independent eigenvectors, that is, 3 linearly independent eigenvalues

[tex]det(A-xI)=det

\begin{pmatrix}

5-x & -2 & 4\\

0 & 3-x & t\\

0 & 0 & 5-x

\end{pmatrix}

[/tex]

[tex]=(5-x)det

\begin{pmatrix}

3-x & t\\

0 & 5-x

\end{pmatrix}

[/tex]

[tex]= (5-x)((3-x)(5-x)-0t)[/tex]

The eigenvalues are 3 and 5.

Obviously, it doesn't matter what t is, we will not be able to get the matrix A to be diagonalizable.

My professor said that he thought there was one correct value for t (but he wasn't sure). Is what I've done correct?