Give a geometric description of the matrix transformation $f: R^2 \rightarrow R^2$ defined by $f(\mathbf{u})=A \mathbf{u}$ for the given matrix $A$.

(a)

$$A=\left[\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right]$$

(b)

$$A=\left[\begin{array}{rr}0 & -1 \\ -1 & 0\end{array}\right]$$

Find a $2 \times 2$ matrix B $\neq$ O and B $\neq$I$_2$ such that AB = BA, where

$$A= \begin{bmatrix} 1&2\\2 & 2\end{bmatrix}$$

. How many such matrices B are there?

If possible, find the inverse of the matrix

$$\left[\begin{array}{lll} 1 & 2 & 3 \\ 2 & 5 & 3 \\ 1 & 0 & 8 \end{array}\right] \text {. }$$

Given the linear system

$$\begin{align*} 2x-y&= 5\\ 4x-2y&=t, \end{align*}$$

(a) Determine a particular value of $t$ so that the system is consistent.

(b) Determine a particular value of $t$ so that the system is inconsistent.

(c) How many different values of $t$ can be selected in part (b)?

Give a geometric description of the matrix transformation $f: R^2 \rightarrow R^2$ defined by $f(\mathbf{u})=A \mathbf{u}$ for the given matrix $A$.

(a)

$$A=\left[\begin{array}{rr}-1 & 0 \\ 0 & 1\end{array}\right]$$

(b)

$$A=\left[\begin{array}{rr}0 & -1 \\ 1 & 0\end{array}\right]$$

Determine the number of entries on or above the main diagonal of a $k \times k$ matrix when (a) $k=2$; (b) $k=3$; (c) $k=4 ;$ (d) $k=n$.

Let the $n \times n$ matrix A be equal to A$^T$. Briefly describe the pattern of the entries in A.

This exercise involves bit matrices.

How many $n \times n$ bit matrices are there?

Consider the following matrixes:

$$A=\begin{bmatrix} 1 & 2 & 3\\ 2 & 1& 4 \end{bmatrix}$$

,

$$B=\begin{bmatrix} 1 & 0 \\ 2 & 1 \\ 3 &2 \\ \end{bmatrix}$$

,

$$C=\begin{bmatrix} 3 & -1 & 3\\ 4 & 1 & 5\\ 2 & 1 & 3\\ \end{bmatrix}$$

,

$$D=\begin{bmatrix} 3 & -2 \\ 2 & 5 \end{bmatrix}$$

,

$$E=\begin{bmatrix} 2 & -4 & 5\\ 0 & 1 & 4\\ 3 & 2 &1\\ \end{bmatrix}$$

,

$$F=\begin{bmatrix} -1 & 2 \\ 0 & 4 \\ 3 & 5 \\ \end{bmatrix}$$

If possible. compute the following:
(a) $A(BD)$
(b) $(AB)D$
(c) $A(C+ E)$
(d) $AC + AE$
(e) $(2AB )^T$ and $2(AB)^T$
(f) $A(C - 3E )$

(a) Show that every scalar matrix is symmetric. (b) Is every scalar matrix nonsingular? Explain. (c) Is every diagonal matrix a scalar matrix? Explain.

Let

$$I_2=\begin{bmatrix} 1 & 0 \\ 0 & 1 \\ \end{bmatrix}$$

and

$$D=\begin{bmatrix} 2 & 3 \\ -1 & 2 \end{bmatrix}$$

, compute $DI_2$ and $I_2D$.

Explain why every incidence matrix A associated with the graph is the same as A$^T$.

Show that a matrix that has a row or column consisting entirely of zeros must be singular.

If $B=P A P^{-1}$, express $B^2, B^3, \ldots, B^k$, where $k$ is a positive integer, in terms of $A, P$, and $P^{-1}$.