QUESTION $1$

$($a$)$ Suppose that $A$ is a $22$ matrix with real numbers as elements:

$A=\left[\begin{array}{cc}\frac{e^6+1}{3e}& \frac{e^6-1}{3e}\\ \frac{e^6-1}{6e}& \frac{e^6+1}{3e}\end{array}\right].$

Compute $($i$)-($iv$)$ and provide the supporting code by using Octave$/$MATLAB$.$

$($i$)$ Inverse of A$.$

$($ii$)$ Transpose of $A^{-1}.$

$($iii$)$ Trace of $A^{-1}.$

$($iv$)$ Determinant of $A.$

$($b$)$ The prices of three commodities gold $($G$),$ silver $($S$)$ and platinum $($P$)$ are in Rands $x,y$ and $z$ per units respectively. Person A purchases $4$ units of $S$ and sells two units of $G$ and $5$ units of $P.$ Person B purchases $2$ units of P and sells $3$ units of G and one unit of S$.$ Person C purchases one unit of G and sells $3$ unit of S and one unit of P$.$ In the process, A$,$ B and C earn R$15000,$ R$1000$ and R$4000$ respectively. Determine the prices per unit of G$,$ S and P by using the $\frac{A}{\frac{?}{?}}b$ construct method. $($Hint: Use GNU Octave to solve a system of three equations.$)$

$($c$)$ Suppose that a banker wants to code a system to calculate the compound interest. To build this model we consider the principal amount A that is invested for a term of $n$ years, with an annual interest rate that grows at $A(1+r{)}^n.$ The banker wants to know the final balance for the investments of R$750,$ R$1000,$ R$3000,$ R$5000,$ and R$11999$ in his portfolio over $10$ years with interest rate of $9\%.$

Write an Octave$/$Matlab code that impliments and solves the above problem.

$($d$)$ The steady$-$state current I flowing in a circuit that contains a resistance $R=5,$ capacitance $C=10,$ and inductnce $L=4$ in series is given by:

$I=\frac{E}{\sqrt[\phantom{2}]{R^2+(2L-\frac{1}{2C}{)}^2}}$

where $E=2$ and $=2$ are the input voltage and angular frequency, respectively. Write a MATLAB$/$Octave program that computes the value of I.