Here is another problem with approximate computer arithmetic that we will encounter in the next section. Consider the matrix

\begin{equation*} A = \left[\begin{array}{rrr} 0.2 \amp 0.2 \amp 0.4 \\ 0.2 \amp 0.3 \amp 0.1 \\ 0.6 \amp 0.5 \amp 0.5 \\ \end{array}\right]\text{.} \end{equation*}
  1. Notice that this is a positive stochastic matrix. What do we know about the eigenvalues of this matrix?

  2. Use Sage to define the matrix \(A\) and the \(3\times3\) identity matrix \(I\text{.}\) Ask Sage to compute \(B = A-I\) and find the reduced row echelon form of \(B\text{.}\)

  3. Why is the computation that Sage performed incorrect?

  4. Explain why using a computer to find the eigenvectors of a matrix \(A\) by finding a basis for \(\nul(A-\lambda I)\) is problematic.