TweetFollow Us on Twitter

Volume Number:4
Issue Number:9
Column Tag:Basic School

Matrices in True Basic

By Dave Kelly, MacTutor Editorial Board

Using Cramer’s Rule and True Basic to Solve Simultaneous Equations

It is only a matter of time until each Engineering or Math student comes across the need to solve simultaneous equations. Fortunately, we live in a time when someone has already figured out the solutions and we can readily use the methods which have been discovered. In this column I hope to show how True Basic’s MAT statements can be used with Cramer’s Rule to solve simultaneous equations.

In most areas of engineering, the time comes from time to time to solve equations. Many of these solutions can be found using a pencil and paper and a little bit of logical thinking. We learn from algebra that to solve a set of equations, there needs to be one equation generated for each unknown variable. It is the responsibility of the engineer to use his expertise in his field of study to determine what the equations are. The solution of the equations are sometimes trivial, but may take time to manipulate the equations with simple algebra. Eventually the answer is found.

Computers are supposed to make life easier. Right? Well, of course! So let’s see how we can solve some equations using True Basic. First, we need a problem to solve: We want to find the current in each of three paths of a circuit. The circuit is shown in figure 1.

To determine the equations which we will use to solve the problem we will use Ohm’s law and Kirchhoff’s voltage law. For the non-engineer, Ohm’s law is a mathematical way to express the relationship between voltage, current, and resistance:

E = I * R

Figure 1.

where R is resistance, E is the voltage across the resistance, I is the current through the resistance. This simple law is basic to a beginning electronics class. Ohm’s law is named after a German physicist, George Simon Ohm, who published a pamphlet in 1827 which contained the results of his efforts to describe and relate currents and voltages mathematically. It has since been shown, however, that this result was discovered 46 years earlier in England by Henry Cavendish.

Kirchhoff’s voltage law is named after Gustav Robert Kirchhoff, a German university professor who was born about the time Ohm was doing his experimental work. Kirchhoff’s law states that the algebraic sum of the voltages around any closed path in a circuit is zero. I won’t attempt to prove this as there are many electronics books which explain how Kirchhoff’s voltage law works.

Figure 2.

Our first equation is derived by taking the loop shown in Figure 2. The path of the loop does is not important as long as the path is closed. The equation is:

-7 + 1(i1 - i2) + 3 (i3 - i2) + 1i3 = 0

which equates to:

i1 - 4i2 + 4i3 = 7.

Our second equation is derived by taking the loop where current i2 is as shown in Figure 1. The equation is:

1(i2 - i1 ) + 2 i2  + 3(i2 - i3) = 0

which equates to:

-i1 - 6i2 - 3i3 = 0.

The last equation comes from the relationship of the current source and the currents i1 and i2:

i1 - i3  =  7.

Now we have three equations and three unknown values to solve for. An easy method to use is given by Cramer’s Rule. Gabriel Cramer (1704-1752), was a Swiss mathematician, also known by his book on the theory of curves, which appeared in 1750 in Geneva. Cramer’s Rule states that if we have a system of linear equations

a11x1 + a12x2 + + a1nxn = b1,
a21x1 + a22x2 + + a2nxn = b2,
an1x1 + an2x2 + + annxn = bn

with n equations and n unknowns., the n X n coefficient matrix of the system is denoted by A. The determinant A 0, the components of the solution of the system are given by the formula,

where the matrix Bk is the same as A except that the elements aik, 1 ¾ i ¾ n, in the kth column of A have been replace by the terms bi, 1 ¾ i ¾ n, respectively.

Cramer’s Rule says that n simultaneous equations with n unknowns can be determined by taking determinants of the matrix. Each of the unknowns is calculated as follows:





The great thing about this is that with True Basic the solution is easy to calculate because of the MAT statements included in True Basic. MAT statements include MAT INPUT (for inputing an array), MAT LINE INPUT, MAT PLOT (plotting to a True Basic chart), MAT PRINT (to print a matrix, MAT READ (read an array from data statements), MAT REDIM (to reset dimensions of a matrix), and MAT WRITE (to write an array to a file). The MAT statement is an enhancement to Basic that is very welcome in the scientific/engineering community. It has been included for many years in Hewlett Packard Basic (now version 5.0).

Using Cramer’s Rule by hand the evaluation of the determinant for i3 is:

evaluation of the other determinants gives i1= 9 Amps and i2= 2.5 Amps. The evaluation of these determinants are fairly easy by hand, but with more equations and more unknowns, the it gets to be a lot of number crunching. The program at the end will do the number crunching for an n X n matrix, limited by the amount of memory you have. The number of equations is calculated in a function which uses the factorial function included in the True Basic Libraries to figure how much data is available in the DATA statements. The coefficients are entered into the DATA statements with equation coefficients in the first DATA statement and the solution coefficients in the second DATA statement. For solving for a large number of equations/unknowns you may break up the DATA into multiple lines.

Once the number of equations is determined, the program calls the sim_solve subroutine to solve the equation. Since the MAT statement does all the work, the subroutine is short and after a couple of MAT assignment statements, the matrix is ready to be crunched. True Basic also supplies us with the det function which will automatically calculate the determinant of an n X n matrix. It sure beats keeping track of a bunch of numbers by hand like I did when I went to school. Calculations are now done and our circuit has been analyzed! Now if they would only put a Mac on my desk I could use it!

! Simultaneous Equation Solver
! Dave Kelly
! ©1988 MacTutor

LIBRARY “Fnmlib”

DEF Get_Equation_Count            !Read the number of equations
         LET Count=0
            READ value
            LET Count=Count+1
    LET n=0
       LET n=n+1
    LOOP UNTIL Factrl(n)>Count
    LET Count=n-1
    LET Get_Equation_Count=Count

! Start Main Program
LET number_of_equations=Get_Equation_Count
DIM a(2,2),b(2,2),c(2,1),x(2)
! Set up all the equations
MAT READ a(number_of_equations,number_of_equations)
MAT READ c(number_of_equations,1)
MAT REDIM x(number_of_equations)
CALL sim_solve(a,b,c,x,number_of_equations)
MAT PRINT x                       ! Print the solutions
DATA 1,-4,4,-1,6,-3,1,0,-1        ! equation coefficients
DATA 7,0,7                        ! equation data

SUB sim_solve(a(,),b(,),c(,),x(),number_of_equations)
    IF det(a)=0 THEN EXIT SUB
    FOR j=1 to number_of_equations
        MAT b=a
        FOR i= 1 to number_of_equations
            LET b(i,j)=c(i,1)
        NEXT i
        LET x(j)=det(b)/det(a)
    NEXT j

Someone told Dave Kelly that he was a programmer. They were kind, but not very accurate. His article “Hierarchical Menus & Colors Notes” [VOL. 4 NO. 6] contained more errors than accuracies. He claims that ZBasic source code must be compiled and the resources attached to make use of them. Why not use the function: “RefNum% = FN OPENRESFILE(FileName$)”?

This makes resources available during the programming phase and requires the deletion of only one command before creating the application.

The next glaring error was his slap-dash program that should have shown how to use submenus from ZBasic.[Excuse me, but I showed A way to to submenus from ZBasic, not THE only way. Actually, I like your way better now that you have shown it to us. Thanks -DK] As the following program demonstrates, submenus do not require special event handling loops, nor is it necessary to append special resources.

‘                    SubMenus from ZBasic
TEXT 0,12,0,0
‘----------------------Set The Menu------------------
MENU 1,0,1,”File”:MENU 1,1,1,”Quit/Q”
SubMenu=150 ‘This is the number we’ll use for our SubMenu
MENU SubMenu,0,1,”I’m outa here”  ‘We’ll delete this from the menu bar 
as soon as we’ve attached it to a menu item.
MENU SubMenu,1,1,”Plain/P;Bold/B<B;Italic/I<I;Outline/O<O;Shadow/S<S”
SubMenuHndl&=FN GETMHANDLE(SubMenu)  ‘Handle to the SubMenu
‘now delete the name from the menu bar
‘CALL DELETEMENU(SubMenu)  ‘(the menu itself still remains available)
MENU 2,0,1,”Format”‘We’ll add the submenu here
‘Create a menu item with the command key equivalent of
‘CHR$(27) and mark the item with the number of the
‘SubMenu instead of a “2” for a check mark
MENU 2,1,SubMenu,MenuName$
‘Use a negative insertion number(-1) and the handle to
‘our SubMenu - the “CALL INSERTMENU(SubMenuHandle,-1)”
‘should take place immediately after the root menu is created
MENU 2,2,1,”Itz Eazy With Z”  ‘Add more items and menus if you like
CALL DRAWMENUBAR ‘Redraw the bar to exclude the hidden menu
ON MENU GOSUB “Handle Menu”
MENU ON ‘We’ll just track menu events
“Loop”  ‘Loop and wait
GOTO “Loop” ‘Getting dizzy?
“Handle Menu”’----------------Menu Handling------------------
MENU  ‘Get results of the menu action
IF MenuID=1 AND ItemID=1 THEN END  ‘End if user selected “Quit”
PRINT@(9,1)MenuID”       “
PRINT@(9,3)ItemID”       “ ‘Else-Show menu info
RETURN  ‘Back to the “Loop”

Wake up, Dave. And try to spend a little time developing your programming skills and a little less time throwing rocks.


Chris Stasny

Well, Chris it seems you have a little chip on your shoulder about something (Do you work for Zedcor?). In my defense, there are many “programmers” out there that have had trouble getting ZBasic event processing to do everything they want it to do. It is true that some of the problems have been fixed as the users have been debugging ZBasic ever since it was released! In my opinion, there is no Basic available today for the Macintosh which is satisfactory for doing serious software development as there are glaring holes in their capabilities. I’ve had too many phone calls from disgruntled developers trying to find a solution to their problems with no solution in sight. Call it rock throwing if you like, but face it, there isn’t any such thing as LightSpeed Basic. Thank you for your letter. We encourage others to share the technology by writing for MacTutor. We are dedicated to the distribution of useful programming information without regard to race, creed or developer status. Chris, we never claim to know everything. We could use more of your good ideas. Write to MacTutor and ask for our authors kit and share some of your “expertise”.

Dave Kelly

I’m a new subscriber to MacTutor and have particularly enjoyed your column “Basic School”. I’m a BASIC programmer and frankly don’t have the time to really learn the other high level languages such as C or Pascal. It is, therefore, heartening to find a source of instruction for BASIC on the MAC.

Do you think it would be feasible to compile your articles into a separate volume? I believe such a book would sell quite well. There are many recreational and small application programmers who both love BASIC and the MAC. Keep up the great work.


Julian Wan

Thank you for your kind remarks. I’ve had several people ask for a separate volume of “Basic School”. Because of the cost of publishing another compiled book and since we already offer the “Best of MacTutor” Vol 1 and 2 for a reasonable price, “Basic School” will not become a volume of its own. Have heart though, I’ve used information from C, Pascal, and assembly language columns many times and so having the other language available in “Best of MacTutor” can actually help when writing BASIC programs too. I’ve also compiled a MacTutor Index HyperCard Stack which can help when you are trying to learn about a specific subject.

Dave Kelly


Community Search:
MacTech Search:

Software Updates via MacUpdate

Between 2 Taps - Tap for Tap interview M...
Hello, and welcome back to Between 2 Taps, Tap for Tap’s Indie Dev interview series. [Read more] | Read more »
Facility 47 (Games)
Facility 47 1.0.1 Device: iOS Universal Category: Games Price: $3.99, Version: 1.0.1 (iTunes) Description: You wake up alone and freezing in an icy cell. You try the cell door but it’s locked, it seems that you are stuck with no... | Read more »
The best Photoshop alternative on iPad
Instagram and Lightroom are great and all, but sometimes people need to get extra creative with their image editing.Like, Photoshop creative. If you're one of these people, take a look at our pick for the best mobile Photoshop experience on iPad... | Read more »
The Walking Dead: No Man’s Land guide -...
A new update for The Walking Dead: No Man’s Land was released last week, making it the perfect time for you to head back to your base and take out some walkers. Here’s the lowdown on what’s new to the game, and how to take advantage. [Read more] | Read more »
Goat Rider guide - Tips and tricks to st...
We've all been there. One second, we're riding high on a crazed goat, and the next, we've been tossed off it like someone who's no good at goat ridin'. [Read more] | Read more »
Real Boxing 2 CREED: How to become a gre...
Just in time for Rocky fans who can’t wait to see CREED, the latest movie, we have the official tie-in game,Real Boxing 2 CREED. It builds on the success of its predecessor and there’s lots to take in so we at 148apps thought we’d run you through... | Read more »
CoinOp Heroes 2 guide - How to build an...
CoinOp Heroes 2 justlaunched and, like all clickers, it's dangerously addictive stuff. You have to furiously tap your screen to defeat wave after wave of foes and earn an insane amount of cash to spend on character upgrades and an army of minions... | Read more »
Dr. Panda Firefighters (Education)
Dr. Panda Firefighters 1.0.1 Device: iOS Universal Category: Education Price: $2.99, Version: 1.0.1 (iTunes) Description: FIGHT FIRES AND SAVE THE DAY!Work together with Dr. Panda and his firefighting team to rescue his trapped... | Read more »
Puddle + (Games)
Puddle + 1.0 Device: iOS iPhone Category: Games Price: $2.99, Version: 1.0 (iTunes) Description: Puddle is back in a new "+" edition featuring enhanced graphics, new videos and Apple TV support ! No IAP and No Ads. Dive into Puddle... | Read more »
Football Manager Mobile 2016 (Games)
Football Manager Mobile 2016 7.0.0 Device: iOS Universal Category: Games Price: $8.99, Version: 7.0.0 (iTunes) Description: Football Manager Mobile 2016 is designed to be played on the move and is the quickest way to manage your... | Read more »

Price Scanner via

Adorama Black Friday deals: Up to $400 off Ma...
Adorama has released their Black Friday deals for 2015. Save up to $400 on MacBook Pros, $200 on MacBooks and MacBook Airs, and $270 on iMacs. Use code RYBFDEAL during checkout to see these prices.... Read more
B&H Photo Deals: $200 off 12-inch 1.2GHz...
In addition to the B&H Photo Black Friday week sales we posted yesterday, B&H has lowered their price on two products to $200 off MSRP: - 12″ 1.2GHz Gray Retina MacBook: $1399 save $200 - 13... Read more
Best Buy Early Access: Today only, Up to $125...
Best Buy has iPad Air 2s on sale for up to $125 off MSRP and Apple Watch models on sale on their online store for up to $100 off MSRP with special codes through midnight CT tonight. Choose free... Read more
UPPERCASE DESIGNS Premium Ultra Thin Keyboard...
UPPERCASE Designs today announced its new Premium Ultra Thin Keyboard Protector and its Palm Rest Protector Set for the 12-inch MacBook. The accessories provide durable protection for the 12-inch... Read more
Al Jazeera Launches New iOS And Android Mobil...
Doha, Qatar based Al Jazeera has launched new mobile and tablet apps on the iOS and Android systems bringing the latest Al Jazeera news and programmes live together with on-demand personalisation.... Read more
B&H Photo Holiday Sale: Up to $250 off Ma...
B&H Photo has all new Macs on sale for up to $500 off MSRP as part of their Holiday sale including free shipping plus NY sales tax only: - 15″ 2.2GHz Retina MacBook Pro: $1799 $200 off - 15″ 2.... Read more
Free Aura ‘Ultimate’ Mac App For Gmail Update...
Miami, Florida based Crosscoded has announced Aura 1.2.0, an update to the Mac app for Gmail. Aura mixes the power of a native client with the flexibility of the Gmail web app with support for up to... Read more
Apple Will Edge Closer to Samsung in Smartpho...
Total smartphone shipments for 2015 are projected to decline by 9.7% to 1.286 billion units, according to the latest report from global market research firm TrendForce. Though Chinese vendors have... Read more
Sidefari – Split Screen Multitasking In Safar...
Francisco Cantu’s Sidefari is a simple web browser designed to act as a companion to Safari on the iPad. With multitasking in iOS 9, Sidefari uses the new Safari View Controller to show an extra... Read more
12-inch MacBooks in stock for up to $120 off,...
Adorama has 12″ Retina MacBooks in stock for up to $120 off MSRP including free shipping plus NY & NJ sales tax only. For a limited time, Adorama will include a free Apple USB-C to USB Adapter,... Read more

Jobs Board

*Apple* Retail - Multiple Positions (US) - A...
Sales Specialist - Retail Customer Service and Sales Transform Apple Store visitors into loyal Apple customers. When customers enter the store, you're also the Read more
Merchant Operations Manager: *Apple* Pay -...
Changing the world is all in a day's work at Apple . If you love innovation, here's your chance to make a career of it. You'll work hard. But the job comes with more than Read more
*Apple* Pay QA Manager - Apple Inc. (United...
Changing the world is all in a day's work at Apple . If you love innovation, here's your chance to make a career of it. You'll work hard. But the job comes with more than Read more
Sr Software Engineer *Apple* Pay - Apple In...
Changing the world is all in a day's work at Apple . If you love innovation, here's your chance to make a career of it. You'll work hard. But the job comes with more than Read more
Hardware Systems Architect - *Apple* Watch...
# Hardware Systems Architect - Apple Watch Job Number: 38449977 Santa Clara Valley, Califo ia, United States Posted: Apr. 16, 2015 Weekly Hours: **Job Summary** The Read more
All contents are Copyright 1984-2011 by Xplain Corporation. All rights reserved. Theme designed by Icreon.