a$.$ For fun, verify the following properties numerically for any "random" square matrices $A,B,$ and $C$ of

your own choice. Turn in just one $($and only one$)$ Matlab output for each problem below:

i$.$ If $A$ is symmetric, the matrix $D=AB$ is$,$ in general, not symmetric, even if $B$ is also symmetric.

ii$.$ If $A$ is symmetric, the matrix $D=B^{T}AB$ is always symmetric.

iii. If $AB=CB,$ it does not necessarily follow that $A=C.$

iv$.$ det$(A)={}_j,$ where ${}_j$ is the $j-$th eigenvalue of $A($use Matlab's help to learn how to use det,

prod, and eig$).$

v$.$ The condition number of a matrix is given as $A=||A||||A^{-1}||$ where $||||$ is a given matrix norm

$($use Matlab's help to learn about the commands norm, cond, and inv$).$

b$.$ Now, let's practice how to plot functions in Matlab by solving each of the exercises described below.

Here, you are asked to turn in a snippet of all the Matlab functions you used in addition to the figures

you generated by those pieces of code.

i$.$ Use Matlab's $1$ inespace function to create a vector $x$ of $100$ entries ranging from $x=0$ to $x=$

$2$ and use it to plot the function $f(x)=sinx.$ Define $f(x)$ by creating a handle to an anonymous

function $($look for "Create Function Handle" in Matlab's help$).$ Don't forget to label your $x$ and $y$

axes using xlabel and ylabel. If you don't know how to plot a function in Matlab, look for

plot on Matlab's help.

ii$.$ Use Matlab's meshgrid function to create a uniformly spaced grid of $x$ and $Y$ values, each ranging

from $0$ to $4$ and use Matlab's surf function to plot the surface $f(x,y)=sinxsiny.$ As with

the previous problem, define $f(x,y)$ using a function handle and don't forget to label your axes.

For this exercise, you should show the surface in two different ways: $($a$)$ using the default

surf$(x,Y,Z)$ function, and $($b$)$ altering the appearance of the surface, such that the option

of the surface from part $($b$),$ use the command camlight after you plot the surface $($search for

the terms surf and camlight on Matlab's help$).$

iii. Plot the contour lines of the Rosenbrock's function $f(x,y)=100(y-x^2{)}^2+(1-x{)}^2$ and

evaluate $Z=f(x,Y),$ where $x$ and $Y$ are obtained using the meshgrid function with $x$ ranging

from $-2$ and $2$ and $Y$ ranging from $-2$ and $5($search for the cont our function on Matlab's help$).$