Questions

1. You are given that \(-1\) is an eigenvalue of \(\begin{bmatrix} -3 & 4 \\ -1 & 1\end{bmatrix}\). What is the algebraic multiplicity of this eigenvalue?

   The answer is 2.

   The characteristic polynomial of the given matrix is \[\begin{vmatrix} -3 -\lambda & 4\\ -1 & 1-\lambda\end{vmatrix} = (-3-\lambda)(1-\lambda)-4(-1) = \lambda^2 +2\lambda + 1 = (\lambda+1)^2.\] Note that \(-1\) appears as a root to the characteristic polynomial twice. This shows that \(-1\) is an eigenvalue with algebraic multiplicity 2.

2. The matrix \(\begin{bmatrix} 2 & 1 & 0\\ 0 & 2 & 1 \\ 0 & 0 & 2\end{bmatrix}\) has a unique eigenvalue. What is the geometric multiplicity of this eigenvalue?

   The answer is 1.

   Let \(A\) denote the given matrix. The unique eigenvalue of \(A\) is \(2\). The geometric multiplicity of \(2\) is given by the nullity of \(A- 2I = \begin{bmatrix} 0 & 1 & 0\\ 0 & 0 & 1 \\ 0 & 0 & 0\end{bmatrix}\), which is 1 since this matrix is in RREF and has one nonpivot column.