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?

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