Linear Algebra - diagonalizable matrix
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
5 & -2 & 4\\
0 & 3 & t\\
0 & 0 & 5
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
5-x & -2 & 4\\
0 & 3-x & t\\
0 & 0 & 5-x
3-x & t\\
0 & 5-x
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?