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[tex]

A = \left[ {\begin{array}{*{20}c}

7 & { - 2} \\

{15} & { - 4} \\

\end{array}} \right]

[/tex]

I need to diagonalise matrix A. So I need a matrix D such that [tex]D = P^{ - 1} AP[/tex].

I calculate the eigenvalues for A, and got bases for the eigenspace associated with each of the eigenvalues. Following the procedure in my book I took the union of the two(it turned out that there are two bases) bases which I found to be: {(2,5),(1,3)}.

So [tex]P = \left[ {\begin{array}{*{20}c}

2 & 1 \\

5 & 3 \\

\end{array}} \right] \Rightarrow P^{ - 1} = \left[ {\begin{array}{*{20}c}

3 & { - 1} \\

{ - 5} & 2 \\

\end{array}} \right][/tex] where I have formed the matrix P whose columns are the vectors in the set which is the union of the two bases for the eigenspaces.

My calculations yield [tex]D = \left[ {\begin{array}{*{20}c}

2 & 0 \\

0 & 1 \\

\end{array}} \right][/tex].

The answer is [tex]D = \left[ {\begin{array}{*{20}c}

1 & 0 \\

0 & 2 \\

\end{array}} \right][/tex].

I'm not sure where my error is. I've checked the matrix multiplication for D and also PP^-1 = I.