You may recall from third semester algebra Cramer's Rule, named for a Swiss mathematician of the 18th century who first discovered this simple way of solving linear equations where the number of unknowns equals the number of equations (a square matrix). Perhaps because of my affinity for the name, it was part of why I had no problem getting an A in third semester algebra. (Math was the only high school class where I would sometimes get a B -- everything else was really easy.)
I was looking for something else, and I happened on this incredibly cute Russian lady explaining how to apply Cramer's Rule to 3x3 matrices. I fear that if she had been teaching me this technique, instead of Mr. Caballero, my high school math teacher, I would have been too distracted to learn.
Back in engineering school we always just found it easier to solve the 3 simultaneous equations rather than use the "easier" (and it is) but more involved and tedious Cramer's rule.
ReplyDeleteAs you go above 3X3 the difference between the 2 just gets larger, and Cramer's rule (sorry dude, you're great but...) quickly becomes unusable.
Besides Cramer's Rule, the Star Rule for finding the determinant of a given 3x3 matrix is pretty nifty, although I'm not sure if it's all that less work than just using the classic recursive method for finding the determinant.
ReplyDeleteOf course, once you get beyond 3x3 matrices, finding the determinant recursively is computationally intensive in itself.
I don't recall using Cramer's Rule all that much, but I do seem to recall that it's more useful as a theoretical construct, than it is as a practical method of computing things. A lot of things in math are like that! :-)
Well the Ukraine girls really knock me out
ReplyDeleteThey leave the west behind
And Moscow girls make me sing and shout
That Geogie's always on my my my my my my my my my mind
A couple of things.
ReplyDeleteI think it is fairly easy to show, at least on a 2X2 matrix, that Cramer's rule is doing with algebra what you would be doing directly on the numbers of solving the 2 simultaneous equations. I have been taught Cramer's rule (the recursive, not the star form) and have also taught it but never seen it derived, but I suspect what I am suggesting true, that Cramer's rule is doing algebra to get the solution instead of a numerical algorithm.
That being said, if you need a closed-form algebraic formula for solving a system of equations, go to Cramer's rule.
Actually, with my aging brain, if I need a numeric solution for any linear system bigger than 2X2, I go to the LAPACK software library if I am programming in Java, or I use Matlab or any of the free-software clones that are user-friendly interfaces to LAPACK.
If I need an algebraic solution, say, for a scholarly research paper, I go consult what I call "The Electric Graduate Student." Maple. Or the Mathematica software package. Or this free-software package called Sage. Which is probably doing something akin to Cramer's rule behind the scenes.
With these software packages, life is to short to strain to solve anything bigger than 2X2 with the possibility of making math or algebra errors. So if we are using packages, who is going to write the packages? Me. I use packages and also have contributed such packages "to the community."