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Example 1

Show that \(-1\) is an eigenvalue of \(A = \begin{bmatrix} 0 & 2 \\ 1 & 1 \end{bmatrix}\).

Example 2

Show that \(u = \begin{bmatrix} 1\\ -2\end{bmatrix} \) is an eigenvector of \(A = \begin{bmatrix} 2 & 2 \\ 6 & 1 \end{bmatrix}\).

Example 3

Let \(A = \begin{bmatrix} 2 & 2 \\ 1 & -1 \end{bmatrix}\). For what values of \(a\) is \(u = \begin{bmatrix} a\\ -1\end{bmatrix} \) is an eigenvector of \(A\)?

Example 4

You are given that \(2\) is an eigenvalue of \(A = \begin{bmatrix} 2 & 0 & 0 \\ 0 & 3 & -2 \\ 0 & 1 & 0 \\ \end{bmatrix}\). Determine its algebraic and geometric multiplicity.

Example 5

Show that the eigenvalues of a triangular matrix are given by the entries on the diagonal.

Example 6

Let \(A = \begin{bmatrix} 1 & 1 & -1 \\ 0 & 3 & -2 \\ 0 & 1 & 0 \end{bmatrix}.\) You are given that \(1\) is an eigenvalue of \(A\). Give a basis for the eigenspace of \(A\) of this eigenvalue.

Example 7

Let \(A = \left[ {\begin{array}{ccc} 1 & 1 & 0 \\ 0 & 2 & 0 \\ 0 & 2 & 1 \\ \end{array} } \right].\) Diagonalize \(A\) by finding a matrix \(P\) and a diagonal matrix \(D\) such that \(A = PDP^{-1}\).

Example 8

Let \(A\in \mathbb{R}^{m\times n}\). Show that every eigenvalue of \(A^\mathsf{T}A\) is nonnegative.

Example 9

Let \({\cal S}_{n\times n}\) denote the set of \(n\times n\) symmetric matrices with real entries. Show that \({\cal S}_{n\times n}\) form a subspace of \(\mathbb{R}^{n\times n}\).

Example 10

Let \(A = \begin{bmatrix} 10 - 5a & -16 + 10a \\ 5 - 3a & -8 + 6a \end{bmatrix}.\) Determine if there is a value for \(a\) so that \(A\) is not diagonalizable.

Example 11

Let \(A\) be an \(n\times n\) real matrix having \(n\) distinct eigenvalues. Prove that \(\det(A)\) equals the product of the eigenvalues.

Example 12

In diagonalizaling a given matrix \(A \in \mathbb{Z}^{2\times 2}\), it is found that \(A = PDP^{-1}\) with \(P = \begin{bmatrix} 1 & 1 \\ 3 & a\end{bmatrix}\) and \(D = \begin{bmatrix} 1 & 0 \\ 0 & -1\end{bmatrix}\). How many possible integer values are there for \(a\)? (Hint: Use the fact that \(A\) has only integer entries and use the inverse formula to obtain \(P^{-1}\).) not diagonalizable.