1. The problem statement, all variables and given/known data
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]
3. 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(AxI)=det
\begin{pmatrix}
5x & 2 & 4\\
0 & 3x & t\\
0 & 0 & 5x
\end{pmatrix}
[/tex]
[tex]=(5x)det
\begin{pmatrix}
3x & t\\
0 & 5x
\end{pmatrix}
[/tex]
[tex]= (5x)((3x)(5x)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?
