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September 4th, 2017, 01:48 PM   #1
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[Determinants] A -> 2x2. Find all the real values r, x, y.

$\displaystyle A \varepsilon \mu (2x2) , A= \begin{Bmatrix} -2 & 4\\ 4 & -3\end{Bmatrix}$

Find all the real values r, x, y for:

1. $\displaystyle |A-rI|=0 $
2. $\displaystyle (A-rI)\binom{x}{y}=0$

How do I proceed?
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September 6th, 2017, 05:04 AM   #2
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$\displaystyle A= \begin{pmatrix} -2 & 4 \\ 4 & -3\end{pmatrix}$ and $\displaystyle I= \begin{pmatrix}1 & 0 \\ 0 & 1\end{pmatrix}$ so $\displaystyle A- rI= \begin{pmatrix}-2- r & 4 \\ 4 & -3- r\end{pmatrix}$.

What is the determinant of that?

$\displaystyle (A- rI)\begin{pmatrix}x \\ y \end{pmatrix}= \begin{pmatrix}-2- r & 4 \\ 4 & -3- r\end{pmatrix}\begin{pmatrix}x \\ y \end{pmatrix}= \begin{pmatrix}(-2- r)x+ 4y \\ 4x+ (-3- r)y\end{pmatrix}$
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