QUESTION $1$

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

$A=\left[\begin{array}{cc}\frac{e^8+i}{4e}& \frac{e^8-1}{4e}\\ \frac{e^8-t}{8e}& \frac{e^8+1}{4e}\end{array}\right].$

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

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

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

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

$($iv$)$ Determinant of a transpose of $A^{-1}.$

$($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 or 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$10$

$2$

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$)$ Let the national income model be:

$Y=C+I_0+G_0,I_0=1000$

$C=40+0\mathrm{.}8(Y-T),G_0=2800$

$T=700+0\mathrm{.}4Y$

where $Y$ is the national income, $C$ is the consumption expenditure, $I_0$ is the initial investment, $G$ is the government expenditure and lastly $T$ represents the taxes. Solve for $Y,C$ and $T$ using GNU Octave by first rewriting the formula in the form $Ax=b$ and also determine the inverse of matrix A$.$

$($d$)$ The luminous efficiency $($ratio of the energy in the visible spectrum to the total energy$)$ of a black body radiator may be expressed as a percentage by the formulais given by:

$E=64\mathrm{.}77T^{-4}{\displaystyle \underset{3{10}^{-5}}{\overset{8{10}^{-5}}{}}}{x}^{-5}{e}^{(\frac{1\mathrm{.}432}{TJ}-1{)}^{-1}}dx$

where $T=3500k$ is the absolute temperature in degrees Kelvin, x is the wavelength in cm$,$ and the range of integration is over the visible spectrum. Write a GNU Octave program that computes the value of E $.$

$[$TURN OVER$]$