Proof Prove that the matrix $A=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]$ is diagonalizable when $-4 b c<(a-d)^{2}$ and is not diagonalizable when $-4 b c>(a-d)^{2}$ is diagonalizable when $-4 b c<(a-d)^{2}$ and is not diagonalizable when $-4 b c>(a-d)^{2}$
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Step 1
A matrix $A$ is diagonalizable if and only if there exists an invertible matrix $P$ such that $P^{-1}AP$ is a diagonal matrix. In other words, we can find a change of basis that transforms $A$ into a diagonal matrix. Now, let's consider the characteristic Show more…
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Let $A$ be an $n \times n$ matrix, and let $q(A)$ be the matrix $$q(A)=a_{n} A^{n}+a_{n-1} A^{n-1}+\cdots+a_{1} A+a_{0} I_{n}$$ (a) Prove that if $B=P^{-1} A P,$ then $q(B)=P^{-1} q(A) P.$ (b) Prove that if $A$ is diagonalizable, then so is $q(A).$
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Let $$A=\left[\begin{array}{ll}a & b \\c & d\end{array}\right]$$ Show that (a) $A$ is diagonalizable if $(a-d)^{2}+4 b c>0$ (b) $A$ is not diagonalizable if $(a-d)^{2}+4 b c<0$ [Hint: See Exercise 29 of Section 5.1.]
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