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## 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

$$A= \begin{pmatrix} 5 & -2 & 4\\ 0 & 3 & t\\ 0 & 0 & 5 \end{pmatrix}$$

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

$$det(A-xI)=det \begin{pmatrix} 5-x & -2 & 4\\ 0 & 3-x & t\\ 0 & 0 & 5-x \end{pmatrix}$$
$$=(5-x)det \begin{pmatrix} 3-x & t\\ 0 & 5-x \end{pmatrix}$$
$$= (5-x)((3-x)(5-x)-0t)$$

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?

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