My Math Forum [Determinants] A -> 2x2. Find all the real values r, x, y.

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 September 4th, 2017, 12:48 PM #1 Newbie   Joined: Aug 2017 From: Earth Posts: 3 Thanks: 0 [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?
 September 6th, 2017, 04:04 AM #2 Math Team   Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902 $\displaystyle A= \begin{pmatrix} -2 & 4 \\ 4 & -3\end{pmatrix}$ 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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