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.