Math 511: Linear Algebra

Unit III Exam: Review Guide

1. Find the eigenvalues and corresponding eigenspaces for the following matrices.

	a.)	\(\begin{pmatrix} 3 & 2 \\ 4 & 1 \end{pmatrix}\)
	b.)	\(\begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{pmatrix}\)
	c.)	\(\begin{pmatrix} -2 & 0 & 1 \\ 1 & 0 & -1 \\ 0 & 1 & -1 \end{pmatrix}\)

2. Let \(A \in \mathbb{R}^{2\times2}\) with characteristic polynomial \(p(\lambda) = \lambda^2 + b\lambda + c\). Show that \(c = \mathrm{det}(A)\).

3. Find the general solutions of each of the following systems of differential equations.

	a.)	\(\begin{cases} y_1' = y_1 + y_2 \\ y_2' = -2y_1 + 4y_2 \end{cases}\)
	b.)	\(\begin{cases} y_1' = y_1 + y_3 \\ y_2' = 2y_2 + 6y_3 \\ y_3' = y_2 + 3y_3 \end{cases}\)

4. Find the general solution of the system of second order differential equations.

\[\begin{cases} y_1'' = -2y_2 \\ y_2'' = y_1 + 3y_2 \end{cases}\]

5. Factor the following matrices into a product \(A = XDX^{-1}\) where \(D\) is diagonal.

	a.)	\(\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\)
	b.)	\(\begin{pmatrix} 2 & 2 & 1 \\ 0 & 1 & 2 \\ 0 & 0 & -1 \end{pmatrix}\)
	c.)	\(\begin{pmatrix} 1 & 2 & -1 \\ 2 & 4 & -2 \\ 3 & 6 & -3 \end{pmatrix}\)

6. Compute \(e^A\) for each of the matrices in problem 5.

7. Compute the \(e^A\) by applying the definition of the matrix exponential directly.

\[\begin{pmatrix} 1 & 0 & -1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}\]

8. Show that if \(A\) and \(B\) are two \(n \times n\) matrices with the same diagonalizing matrix \(X\), then \(AB = BA\).

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