Question

Suppose that the trace of a 2×2 matrix A is tr(A)=3, and the determinant is det(A)=−40. Find the eigenvalues of A

Answer 1

I think you need to solve below :

\(\large \lambda^2 - \lambda (\text{trace} ) + \text{determinant} = 0 \)

Answer 2

\[
\begin{bmatrix}
a&b\\
c&d
\end{bmatrix}
\]

Answer 3

\[
(a-\lambda)(d-\lambda)-bc=0
\]We know \(ad-bc = -40\) and \(a+d=3\).

Answer 4

Make that \(\lambda_1\) and \(\lambda_2\).

Answer 5

\[
ad-(a+d)\lambda+\lambda^2-bc = 0
\]Then we get: \[
\lambda^2-(\color{red}{a+d})\lambda +(\color{blue}{ad-bc})=0
\]So looking at that, I agree with @ganeshie8

Answer 6

Thank you! I got my small eigenvalue and large eigenvalues mixed up. small eigen.= -5, and large eigen. = 8