Part $3$ Exercises

Recall that $a=$ last digit of your student ID$,b=$ second to last digit of your student ID$,c=$ third to last digit of

your student ID$,$ and $d=$ the sum of the last three digits of your student ID$.$ Complete the following operations in

your script, making new lines for each operation.

$(a+1)x+(b+1)y=-9$

$(c+1)x+(d+1)y=-4$

$3$A$)($Solve system of linear equations, Method $1)$ Solve the above system of linear equations using reduced row

echelon form. In other words, declare a $23$ coefficient matrix called P$3$AM for the system above $($make a matrix

with just numbers, no variables$),$ then use the reduced row echelon form command to solve the system. Declare

the solution in MATLAB as P$3$A$\mathrm{.}3$B$)($Solve system of linear equations, Method $2)$ Solve the same system of linear equations using an inverse

matrix. In other words, declare a $22$ coefficient matrix called P$3$BM for the same system $($make a matrix with

just numbers, no variables, but only for the side with the $x$ and $y$ on it$),$ then declare a $21$ vector called P$3$BV

with the numbers on the right side of the equation. If we let $x=[x$;$y],$ then the system turns into $(P3BM)(x)=$

$(P3BV)$; therefore, the solution to the system should be $x=(P3BM{)}^{-1}(P3BV).$ Use the proper inverse matrix

operation and declare the solution in MATLAB as P$3$B$.$ Check that your results match the prior method from $3$A$.$

$3$C$)($Dot Product $-$ MAT $264)$ Declare P$3$C$1$ as a $12$ row vector with $a+1$ and $b+1$ as the two entries. Declare

$PC_2$ as another $12$ row vector with $c+1$ and $d+1$ as the two entries. Compute the dot product of these vectors

by multiplying the two vectors together, but transpose $PC_2$ before they are multiplied. Declare your product in

MATLAB as P$3$C$.$

$3$D$)($Eigenvalues $-$ MAT $364,$ Method $1)$ You are to solve for the eigenvalues of a $22$ system of differential

equations, given as

$x^{\text{'}}=\left[\begin{array}{cc}a+2& b+2\\ c+2& d+2\end{array}\right]x$

where $x=[x$;$y].$ Declare the coefficient matrix above P$3$DM in MATLAB. You are going to find the

eigenvalues of this matrix by appropriately mimicking the lines of code below in MATLAB, but by using relevant

information and following proper syntax rules. Note that syms allows you to define a variable in MATLAB

symbolically. This command will be used regularly in future projects.

syms $Z$

$z+$det$(,1==0,z)$

$3$E$)($Eigenvalues $-$ MAT $364,$ Method $2)$ You are to find the eigenvalues of the same $22$ system of differential

equations as written above. Declare the coefficient matrix as P$3$EM in MATLAB. You will use the eigenvalue

command from the list on the previous page to identify the eigenvalues of this system. Declare your result in

MATLAB as P$3$E$.$ Check that your results match the prior method from $3$D$.$

Once you have these operations written in proper MATLAB syntax, go to your Editor menu and click Run. If

there are values in your Command Window for all operations without error messages, then your code has

successfully run! If you have error messages, read them carefully and try to resolve them, then click Run again.