Activity4.2.2

The eigenvalues of a square matrix are defined by the condition that there be a nonzero solution to the homogeneous equation \((A-\lambda I)\vvec=\zerovec\text{.}\)

  1. If there is a nonzero solution to the homogeneous equation \((A-\lambda I)\vvec = \zerovec\text{,}\) what can we conclude about the invertibility of the matrix \(A-\lambda I\text{?}\)

  2. If there is a nonzero solution to the homogeneous equation \((A-\lambda I)\vvec = \zerovec\text{,}\) what can we conclude about the determinant \(\det(A-\lambda I)\text{?}\)

  3. Let's consider the matrix

    \begin{equation*} A = \left[\begin{array}{rr} 1 \amp 2 \\ 2 \amp 1 \\ \end{array}\right] \end{equation*}

    from which we construct

    \begin{equation*} A-\lambda I = \left[\begin{array}{rr} 1 \amp 2 \\ 2 \amp 1 \\ \end{array}\right] - \lambda \left[\begin{array}{rr} 1 \amp 0 \\ 0 \amp 1 \\ \end{array}\right] = \left[\begin{array}{rr} 1-\lambda \amp 2 \\ 2 \amp 1-\lambda \\ \end{array}\right] \text{.} \end{equation*}

    Find the determinant \(\det(A-\lambda I)\text{.}\) What kind of equation do you obtain when we set this determinant to zero to obtain \(\det(A-\lambda I) = 0\text{?}\)

  4. Use the determinant you found in the previous part to find the eigenvalues \(\lambda\) by solving \(\det(A-\lambda I) = 0\text{.}\) We considered this matrix in the previous section so we should find the same eigenvalues for \(A\) that we found by reasoning geometrically there.

  5. Consider the matrix \(A = \left[\begin{array}{rr} 2 \amp 1 \\ 0 \amp 2 \\ \end{array}\right]\) and find its eigenvalues by solving the equation \(\det(A-\lambda I) = 0\text{.}\)

  6. Consider the matrix \(A = \left[\begin{array}{rr} 0 \amp -1 \\ 1 \amp 0 \\ \end{array}\right]\) and find its eigenvalues by solving the equation \(\det(A-\lambda I) = 0\text{.}\)

  7. Find the eigenvalues of the triangular matrix \(\left[\begin{array}{rrr} 3 \amp -1 \amp 4 \\ 0 \amp -2 \amp 3 \\ 0 \amp 0 \amp 1 \\ \end{array}\right] \text{.}\) What is generally true about the eigenvalues of a triangular matrix?

in-context