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PirateFan308
#1
Nov24-11, 08:48 PM
P: 94
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(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?
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