Find the characteristic and minimum polynomials of each of the following matrices (a) \(\mid \begin{array}{cc}3 & -1 \mid \\ \mid-1 & 3 \mid\end{array}\) (b) \(\begin{array}{cc}\mid 1 & 1 \mid \\ & \mid 0 & 2 \mid\end{array}\) (c) \(\begin{aligned} \mid 1 &-2 \mid \\ & \mid 0 &-1 \mid \end{aligned}\) (d) \(\begin{array}{rl}\mid 1 & 1 \mid \\ & \mid 0 & 1 \mid\end{array}\) (e) $\begin{array}{rrrr} & \mid 0 & 1 & 0 & 0 \\ & \mid 0 & 0 & 0 & 0 \\ & \mid 0 & 0 & 1 & -2 \mid \\ & \mid 0 & 0 & 1 & -1\end{array}$ (f) $\begin{array}{cccc} & \mid 3 & 1 & 0 & 0 \\ & \mid 0 & 3 & 0 & 0 \\ & \mid 0 & 0 & 2 & 1 \\ & \mid 0 & 0 & 1 & 2\end{array} \mid$

Set the determinant of (A - λI) to 0: \(\begin{vmatrix} 3-\lambda & -1 \\ -1 & 3-\lambda \end{vmatrix} = 0\) Calculate the determinant: \((3-\lambda)(3-\lambda) - (-1)(-1) = \lambda^2 - 6\lambda + 9 - 1 = \lambda^2 - 6\lambda + 8\) The characteristic polynomial is \(p(\lambda)=\lambda^2 - 6\lambda + 8\).

Solve the characteristic polynomial equation \(p(\lambda)=0\): \(\lambda^2 - 6\lambda + 8 = 0\) Factoring yields: \((\lambda - 4)(\lambda - 2) = 0\) Eigenvalues are λ=2 and λ=4.

Since we have two distinct eigenvalues, the minimum polynomial is formed by multiplying the factors of the characteristic polynomial corresponding to each eigenvalue: \(m(\lambda) = (\lambda - 2)(\lambda - 4)\) Thus, the characteristic polynomial for matrix (a) is \(p(\lambda) = \lambda^2 - 6\lambda + 8\) and the minimum polynomial is \(m(\lambda) = (\lambda - 2)(\lambda - 4)\). These explanations are then repeated for each matrix: (b) Matrix: \(\begin{bmatrix} 1 & 1 \\ 0 & 2 \end{bmatrix}\) Characteristic polynomial: \(p(\lambda)=\lambda^2 - 3\lambda + 2\) Minimum polynomial: \(m(\lambda)=(\lambda - 1)(\lambda - 2)\) (c) Matrix: \(\begin{bmatrix} 1 & -2 \\ 0 & -1 \end{bmatrix}\) Characteristic polynomial: \(p(\lambda)=\lambda^2 +\lambda\) Minimum polynomial: \(m(\lambda)=\lambda(\lambda + 1)\) (d) Matrix: \(\begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}\) Characteristic polynomial: \(p(\lambda)=(\lambda - 1)^2\) Minimum polynomial: \(m(\lambda)=(\lambda - 1)^2\) (e) Matrix: \(\begin{bmatrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & -2 \\ 0 & 0 & 1 & -1 \end{bmatrix}\) Characteristic polynomial: \(p(\lambda)=\lambda^3(\lambda - 1)\) Minimum polynomial: \(m(\lambda)=\lambda^2(\lambda - 1)\) (f) Matrix: \(\begin{bmatrix} 3 & 1 & 0 & 0 \\ 0 & 3 & 0 & 0 \\ 0 & 0 & 2 & 1 \\ 0 & 0 & 1 & 2 \end{bmatrix}\) Characteristic polynomial: \(p(\lambda)=(\lambda - 3)^2(\lambda - 2)^2\) Minimum polynomial: \(m(\lambda)=(\lambda - 3)(\lambda - 2)\)