Matrix Operations $40$ Points $/10$ per Part

In this exercise, you will first use for loops or while loops to create a matrix, and then you will practice some matrix operations and verify some properties for matrices. You may need the following commands for this exercise.

The command inv$($A$)$returns the inverse of A when A is invertible.

The command A$\text{'}$ returns the transpose of A when all entries in A are real numbers.

You also need to be familiar with both for loops and while loops.

for loop

The for loop executes a group of statements in a loop for a specified number of

times.

for index $=$ initial$\_$val:end$\_$val

statements

end

for index $=$ initial$\_$val:step:end$\_$val

statements

end

Note: The first for loop has an implicit step of $1.$

while loop

The while loop executes an expression and repeats the execution of a group of statements in a loop while the expression is true $($see the example in the handout$).$

while expression

statements

end

Compare for loop $($i$)$ and for loop $($ii$)$ in the template file.

Do you have the same output for matrix A$?$

For an n x n matrix A$,$ calculate the number of FLOPs $($additions$)$ used in each for loop. Express the number of FLOPs in terms of n for each for loop. Be sure to show your work for for loop $($ii$)$ and simplify your final answer.

Which for loop uses fewer FLOPs?

Practice while loops: Rewrite both for loop $($i$)$ and for loop $($ii$)$ using while loops only. $($You should get the same output as part A$.$ If not, check your code in while loops.$)$

Note: DO NOT forget to put semi$-$colons in commands in loops $($causes excessive printing$)!$

Check some properties for matrix multiplications and do the following operations using matrix A from $3$A:

Create a matrix B with size n x $($n$-2)$ whose entries are random integers between $-2$ and $10.$

Try A$*$B and B$*$A by setting them to ABBA.

Note: Only one command will work. Leave the one that works and give a reason why the other does not work.

Create a matrix C with size n x n whose entries are random integers between $-2$ and $10.$

Calculate A$*$C and C$*$A and set them to AC and CA$,$ respectively. Are they the same? Why or why not?

Calculate A$*$eye$($n$)$ and eye$($n$)*$A and set to AI and IA$.$ Are they the same? Why or why not?

Check some properties involving the inverse and transpose of a matrix.

Use matrix A from $3$A and find the inverse of A$,$ inverse$\_$A$,$ using inv$($A$).$ Is A invertible?

Let

$.$ Create matrix D and find the inverse of D using

inv$($D$),$ and

RREF of D $--$ Write the code with exactly two lines using rref to calculate the inverse of D$.$

Calculate the following in MATLAB.

inv$($inv$($D$)),$ does it equal to D$?$ Generalize your observation.

Create matrix

$.$

inv$($D$*$E$),$ inv$($D$)*$inv$($E$),$ inv$($E$)*$inv$($D$),$ which two give you the same result? Finish the generalization.

inv$($D$\text{'}),($inv$($D$))\text{'},$ are they the same? Finish the generalization.