UNIVERSITY EXAMINATIONS UNIVERSITEITSEKSAMENS
OCTOBER/NOVEMBER 2025
APM1513
Applied Linear Algebra
Examiners:
First: MR Y SITHOLE
Second: PROF AS KUBEKA
100 Marks
3 Hours
This is a closed book and online examination which you have to write within 3 hours and submit online
using the link: https://cset.myexams.unisa.ac.za/my/
The examination paper will be available 15 minutes before the actual examination’s time to give you time
to download the paper. After the 3 hours for the writing of the examination, you will have an extra 30
minutes to upload your attempts/answers.
Use of a non-programmable pocket calculator is allowed.
This web-based examination remains the property of the University of South Africa and may not be
distributed from the Unisa platform. Students who have not utilised invigilation or proctoring tools
will be deemed to have transgressed Unisa’s examination rules and will have their marks withheld.
This examination allows single PDF attachment only as part of your submission.
Answer All Questions and Submit within the stipulated timeframe.
ALL CALCULATIONS MUST BE SHOWN.
This examination question paper consists of 5 pages.
Open Rubric
, 2 APM1513
Oct/Nov 2025
QUESTION 1
(a) Suppose that A is a 3 × 3 matrix with real non-negative elements:
2 3
8 1 6
A=6 7
43 5 75 .
4 9 2
Compute (i)-(iv) and provide the supporting code by using Octave/MATLAB.
(i) Inverse of A. (5)
(ii) Transpose of A—1. (3)
(iii) Trace of A. (1)
(iv) Determinant of A. (1)
(b) 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 R750, R1000, R3000, R5000, and R11999 in his portfolio over 10 years with interest rate of 9%.
Write an Octave/Matlab code that impliments and solves the above problem. (5)
(c) Plot the graph, with a mesh, p
sin x2 + y2
z= p
x2 + y2
in the range —7 ≤ x ≤ 7, —11 ≤ y ≤ 11. (5)
(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:
E
I= q 1
R2 + (2πωL — 2πuC )
2
where E = 2 and ω = 2 are the input voltage and angular frequency, respectively. Write a
MATLAB/Octave program that computes the value of I. (5)
[25]
[TURN OVER]
, 3 APM1513
Oct/Nov 2025
QUESTION 2
(a) Write a MATLAB/Octave program that solves the following system:
10x + 7y + 8z + 7w = 32,
7x + 5y + 6z + 5w = 23,
8x + 6y + 10z + 9w = 33,
7x + 5y + 9z + 10w = 31.
Using the left division operator, compute the residual, the determinant and the conditional es-
timator. Also state whether the system is ill-conditioned or well conditioned and justify your
answer. (15)
(b) Write a program to compute the sum of the series 12 + 22 + 32 .. . such that the sum is as large as
possible without exceeding 100. The program should display how many terms are used in the sum.
. (10)
[25]
QUESTION 3
(a) Write an Octave/MATLAB code that evaluates
Z 3 1 dv.
(1 + v)1.8
1
(3)
(b) The binomial coefficient is widely used in Mathematics and Statistics. It id defined as the number
of ways of choosing r objects out of n without regard to order and is given by
!
n n!
r = r!(n — r)! .
Write a MATLAB/Octave code that computes the binomial coefficient when n = 10 and r = 3.
(5)
(c) Write a simple Octave/MATLAB command that will find both the eigenvalues and eigenvectors of
matrix 2 3
8 1 6
A=6 43 5 75 .
7
4 9 2
(7)
[TURN OVER]
, 4 APM1513
Oct/Nov 2025
(d) Draw a graph of the population of the USA from 1790 to 2000, utilizing the (logistic) model
197273000
P (t) =
1 + e—0.03134(t—1913.25)
where t is the date in years. The actual data (in 1000s) for every decade from 1790 to 1950 is as
follows:
Year Data
1790 3929
1800 5308
1810 7240
1820 9638
1830 12866
1840 17069
1850 23192
1860 31443
1870 38558
1880 50156
1890 62948
1900 75995
1910 91972
1920 105711
1930 122775
1940 131669
1950 150697
Superimpose the data on the graph of P (t). (10)
[25]
OCTOBER/NOVEMBER 2025
APM1513
Applied Linear Algebra
Examiners:
First: MR Y SITHOLE
Second: PROF AS KUBEKA
100 Marks
3 Hours
This is a closed book and online examination which you have to write within 3 hours and submit online
using the link: https://cset.myexams.unisa.ac.za/my/
The examination paper will be available 15 minutes before the actual examination’s time to give you time
to download the paper. After the 3 hours for the writing of the examination, you will have an extra 30
minutes to upload your attempts/answers.
Use of a non-programmable pocket calculator is allowed.
This web-based examination remains the property of the University of South Africa and may not be
distributed from the Unisa platform. Students who have not utilised invigilation or proctoring tools
will be deemed to have transgressed Unisa’s examination rules and will have their marks withheld.
This examination allows single PDF attachment only as part of your submission.
Answer All Questions and Submit within the stipulated timeframe.
ALL CALCULATIONS MUST BE SHOWN.
This examination question paper consists of 5 pages.
Open Rubric
, 2 APM1513
Oct/Nov 2025
QUESTION 1
(a) Suppose that A is a 3 × 3 matrix with real non-negative elements:
2 3
8 1 6
A=6 7
43 5 75 .
4 9 2
Compute (i)-(iv) and provide the supporting code by using Octave/MATLAB.
(i) Inverse of A. (5)
(ii) Transpose of A—1. (3)
(iii) Trace of A. (1)
(iv) Determinant of A. (1)
(b) 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 R750, R1000, R3000, R5000, and R11999 in his portfolio over 10 years with interest rate of 9%.
Write an Octave/Matlab code that impliments and solves the above problem. (5)
(c) Plot the graph, with a mesh, p
sin x2 + y2
z= p
x2 + y2
in the range —7 ≤ x ≤ 7, —11 ≤ y ≤ 11. (5)
(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:
E
I= q 1
R2 + (2πωL — 2πuC )
2
where E = 2 and ω = 2 are the input voltage and angular frequency, respectively. Write a
MATLAB/Octave program that computes the value of I. (5)
[25]
[TURN OVER]
, 3 APM1513
Oct/Nov 2025
QUESTION 2
(a) Write a MATLAB/Octave program that solves the following system:
10x + 7y + 8z + 7w = 32,
7x + 5y + 6z + 5w = 23,
8x + 6y + 10z + 9w = 33,
7x + 5y + 9z + 10w = 31.
Using the left division operator, compute the residual, the determinant and the conditional es-
timator. Also state whether the system is ill-conditioned or well conditioned and justify your
answer. (15)
(b) Write a program to compute the sum of the series 12 + 22 + 32 .. . such that the sum is as large as
possible without exceeding 100. The program should display how many terms are used in the sum.
. (10)
[25]
QUESTION 3
(a) Write an Octave/MATLAB code that evaluates
Z 3 1 dv.
(1 + v)1.8
1
(3)
(b) The binomial coefficient is widely used in Mathematics and Statistics. It id defined as the number
of ways of choosing r objects out of n without regard to order and is given by
!
n n!
r = r!(n — r)! .
Write a MATLAB/Octave code that computes the binomial coefficient when n = 10 and r = 3.
(5)
(c) Write a simple Octave/MATLAB command that will find both the eigenvalues and eigenvectors of
matrix 2 3
8 1 6
A=6 43 5 75 .
7
4 9 2
(7)
[TURN OVER]
, 4 APM1513
Oct/Nov 2025
(d) Draw a graph of the population of the USA from 1790 to 2000, utilizing the (logistic) model
197273000
P (t) =
1 + e—0.03134(t—1913.25)
where t is the date in years. The actual data (in 1000s) for every decade from 1790 to 1950 is as
follows:
Year Data
1790 3929
1800 5308
1810 7240
1820 9638
1830 12866
1840 17069
1850 23192
1860 31443
1870 38558
1880 50156
1890 62948
1900 75995
1910 91972
1920 105711
1930 122775
1940 131669
1950 150697
Superimpose the data on the graph of P (t). (10)
[25]
