(1) Find the characteristic polynomial of the matrix A:

\[ \left[ {\begin{array}{cc} 1 & 2 \\ 3 & 4 \\ \end{array} } \right] \]

Christian’s Response:

Equation to get characteristic polynomial:

\(det(A-\lambda\mathit{I}_n)\)

Solve for the following

\[ \left[ {\begin{array}{cc} 1 & 2 \\ 3 & 4 \\ \end{array} } \right] - \left[ {\begin{array}{cc} \lambda & 0 \\ 0 & \lambda \\ \end{array} } \right] \]

Subtract A and \(\lambda\)

\[ \left[ {\begin{array}{cc} 1-\lambda & 2 \\ 3 & 4-\lambda \\ \end{array} } \right] \]

Breakdown equation

\((1-\lambda)(4-\lambda)-6\)

Characteristic polynomial below

\(\lambda^2-5\lambda-2\)