Solve any maths problem

Step by step

1. \det(A - \lambda I) = 0

   Eigenvalues are the roots of the characteristic polynomial.

2. \det\left[\begin{matrix}- \lambda & 0\\0 & 1 - \lambda\end{matrix}\right] = 0

   Subtract λ from the diagonal.

3. \lambda^{2} - \lambda = 0

   Expand the determinant.

4. \lambda \left(\lambda - 1\right) = 0

   Factor.

5. \lambda = 0, \lambda = 1

   Eigenvalues (with multiplicity).

6. \lambda = 0:\ (A - \lambda I)\mathbf{v} = 0 \Rightarrow \mathbf{v} = \left[\begin{matrix}1\\0\end{matrix}\right]

   Solve (A − 0I)v = 0 for a basis eigenvector.

7. \lambda = 1:\ (A - \lambda I)\mathbf{v} = 0 \Rightarrow \mathbf{v} = \left[\begin{matrix}0\\1\end{matrix}\right]

   Solve (A − 1I)v = 0 for a basis eigenvector.