Exercise L$1$

$[20$pts$]$ Consider the following matrices $A,B,$ and $C,$ and use them in all parts below.

$A=([6,-5,-2],[2,-4,5],[0,1,-8])$;$B=([2,3,5],[-1,1,6],[8,-5,-3])$;$C=([-1,-4,1],[2,2,0],[-4,3,2])$

$($a$)$ What is $AB?$ What is $BA?$ Show that $ABBA.$

$($b$)$ Show that $(AB)C=A(BC).$

$($c$)$ Show that $(A+B)C=AC+BC.$

$($d$)$ Show that $(AB{)}^T=B^T{A}^T.$ Show that $(AB{)}^T{A}^T{B}^T.$

$($e$)$ What is $A^{-1}?$ What is $B^{-1}?$ What is $(AB{)}^{-1}?$

Show that $(AB{)}^{-1}{A}^{-1}{B}^{-1}.$

Show that $(AB{)}^{-1}=B^{-1}{A}^{-1}.$

$($f$)$ What is det$(A)?$ What is det$(B)?$ What is det$(AB)?$

Show that det$(AB)=$det$(A)det(B).$

$($g$)$ Show that det$(1\mathrm{.}2A)=(1\mathrm{.}2{)}^3$det$(A).$

NOTES:

Write a MATLAB script to answer these questions. Be sure to include your name, a date, the

assignment, and a description.

Your script should be efficient. This means you should define variables that you are using more than

once, e$.$g$.,$ the matrix $AB$ should be defined, as it can be used several times.

Note that the best way to "show that" something is true or not true is to construct an expression that

is output to the Command Window. For instance, for part $($b$),$ rearrange the equation so that

something is equal to zero, compute that expression, and an output of an array of $0$ s is sufficient to

show that the left side is equal to the right side. DISP commands are not needed, especially if you

choose meaningful variable names. Logical expressions are also not needed.

If possible, write each check in one expression, i$.$e$.,$ you never need to define something, then use it

once. For example, once $AB$ and $BC$ are defined, you can write the checks for parts $($b$)$ and $($c$)$ in one

expression each.

On the other hand, if you need something multiple times, define it then use it$.$ For example, in part

$($d$),$ you will use $B^T,A^T,$ and $(AB{)}^T$ twice each, so you should define each before writing the checks.

Comments should be specific, telling the reader exactly what you expect the result of each check to be$.$

For instance, if you expect a $33$ array of $0$ s$,$ then that's what you should write as your comment.

The output needs to be meaningful, with meaningful variable names, e$.$g$.,$ check$\_$a for the first check.

Suppress output for any line of code that does not specifically answer one of the questions above. For

instance, you do not need to output $($AB$){?}^T$ in part $($d$).$

Combine the MATLAB script and output from the Command Window into a single PDF before you

submit it to Moodle. Clear the Command Window before your last run.