Due on Wednesday, Feb 6.
Show that for any \(A \in M_2\), the characteristic polynomial of \(A\) is \(t^2 - \operatorname{tr}(A) t + \det(A)\). Show that this implies that \(\operatorname{tr}(A) = \lambda_1 + \lambda_2\) and \(\det(A) = \lambda_1 \lambda_2\), where \(\lambda_1\) and \(\lambda_2\) are the eigenvalues of \(A\).
For matrices \[A = \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix} \text{ and } B = \begin{bmatrix} b_{11} & b_{12} & b_{13} \\ b_{21} & b_{22} & b_{23} \\ b_{31} & b_{32} & b_{33} \end{bmatrix}\] show by direct computation that \(\operatorname{tr}(AB) = \operatorname{tr}(BA)\). Next week we will give a much nicer proof of this fact for any size square matrices.
Show that the eigenvalues of an upper triangular matrix \[T = \begin{bmatrix} t_{11} & \ldots & t_{1n} \\ & \ddots & \vdots \\ \mathbf{0} & & t_{nn} \end{bmatrix}\] are the entries on the main diagonal \(t_{11}, t_{22}, \ldots, t_{nn}\).
Use Theorem 1.2.12 to find the characteristic polynomial of \[\begin{bmatrix} 1 & 1 & 0 & 0 \\ 1 & 1 & 1 & 0 \\ 0 & 1 & 1 & 1 \\ 0 & 0 & 1 & 1 \end{bmatrix}.\]
Explicitly compute the \(k^\text{th}\) elementary symmetric functions \(S_k(\lambda_1, \lambda_2, \lambda_3, \lambda_4)\) for \(k = 1, 2, 3,\) and \(4\).