If $A$ is an $n \times n$ matrix with every entry an odd integer, explain why $\operatorname{det} A$ must be divisible by $2^{n-1}$.

Assume that $A$ is a square matrix and the sum of the entries in each column is $0 .$ Explain why $\operatorname{det} A=0$.

Consider the sequence of numbers defined recursively by

$$a_0=-3, a_1=-2 \text { and, for } n \geq 1, a_{n+1}=5 a_n-6 a_{n-1} \text {. }$$

(a) List the first eight terms of this sequence.

(b) Determine a general formula for $a_n$.

Assume that $A$ and $B$ are similar matrices, then show that $A^2$ and $B^2$ are also similar.

Consider $E$ to be the $n \times n$ elementary matrix obtained by interchanging rows $i$ and $j$ of the $n \times n$ identity matrix $I$. Let $A$ be any $n \times n$ matrix. Determine $\operatorname{det}(E A)$.

Determine eigenvalues of

$$\left[\begin{array}{lll}2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2\end{array}\right]$$

? What are the corresponding eigenspaces?

Given that the eigenvalues of a $3 \times 3$ matrix $A$ are $-1$ and $2,$ what are the possibilities for the characteristic polynomial of $A$?

Consider

$$A=\left[\begin{array}{rrrr}1 & 2 & 3 & 4 \\ 0 & -1 & 1 & 2 \\ 1 & 2 & 3 & 4 \\ 8 & -2 & 5 & 5\end{array}\right]$$

. Calculate $\operatorname{det} A$ by inspection and explain.

(a) Determine the determinant of

$$A=\left[\begin{array}{rrr}-4 & 2 & 1 \\ 0 & -3 & 5 \\ 6 & 4 & -1\end{array}\right]$$

with a Laplace expansion down the first column.

(b) Determine the determinant of $A$ with a Laplace expansion along the third row.

(a) Show that

$$A=\left[\begin{array}{rrr}9 & -\frac{3}{2} & -5 \\ -5 & 1 & 3 \\ -2 & \frac{1}{2} & 1\end{array}\right]$$

is invertible with inverse

$$B=\left[\begin{array}{rrr}1 & 2 & -1 \\ 2 & 2 & 4 \\ 1 & 3 & -3\end{array}\right]$$

.

(b) Find $\operatorname{det} A, \operatorname{det} B$, and $\operatorname{det} 2 A$.

Explain why a square triangular matrix with no zero entries on its diagonal must be invertible.

Assume that $\lambda$ is an eigenvalue of a matrix $A$ and $x$ is a corresponding eigenvector. If $B=P{ }^1 A P$ for some invertible matrix $P$, show that $\lambda$ is also an eigenvalue of $B$. Determine a corresponding eigenvector.

(a) Determine the characteristic polynomial of

$$A=\left[\begin{array}{rr}-2 & -4 \\ 3 & 6\end{array}\right]$$

and use this to explain why $A$ is similar to a diagonal matrix.

(b) Determine a diagonal matrix $D$ and an invertible matrix $P$ such that $P^{-1} A P=D$.

(c) Determine a diagonal matrix $D$ and an invertible matrix $S$ such that $S^{-1} D S=A$.

State at least three conditions on the rows of a matrix $A$ that guarantee that $A$ is singular.