In regard to eigenvalues and eigenvectors in Linear Algebra, How can I prove that the characteristic equation of a $2 \times 2$ matrix $A$ can be expressed as $$ \lambda^2- tr(A)\lambda + \det(A)=0 \, , $$ where $tr(A)$ is the trace of $A$? Please help I'm not even sure where to begin...
2026-03-28 02:59:37.1774666777
Using Eigenvalues to prove a matrix?
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To calculate the characteristic polynomial, we calculate $$det(A-\lambda I).$$ In this case we have: $$\begin{pmatrix} a-\lambda & b \\ c& d-\lambda \end{pmatrix}.$$ can you find the determinant of that? After that just keep in mind that the trace is the sum of the diagonals.