Solution Found!
Let Recall from Exercise 25 in Section 5.4 that trA (the
Chapter , Problem 17E(choose chapter or problem)
Let \(A=\left[\begin{array}{ll}a_{11} & a_{12} \\ a_{21} & a_{22}\end{array}\right]\) Recall from Exercise 25 in Section 5.4 that tr A (the trace of A) is the sum of the diagonal entries in A. Show that the characteristic polynomial of A is
\(\lambda^{2}-(\operatorname{tr} A) \lambda+\operatorname{det} A\)
Then show that the eigenvalues of a \(2 \times 2\) matrix A are both real if and only if \(\operatorname{det} A \leq\left(\frac{\operatorname{tr} A}{2}\right)^{2}\)
Questions & Answers
QUESTION:
Let \(A=\left[\begin{array}{ll}a_{11} & a_{12} \\ a_{21} & a_{22}\end{array}\right]\) Recall from Exercise 25 in Section 5.4 that tr A (the trace of A) is the sum of the diagonal entries in A. Show that the characteristic polynomial of A is
\(\lambda^{2}-(\operatorname{tr} A) \lambda+\operatorname{det} A\)
Then show that the eigenvalues of a \(2 \times 2\) matrix A are both real if and only if \(\operatorname{det} A \leq\left(\frac{\operatorname{tr} A}{2}\right)^{2}\)
ANSWER:Solut