When you are done you will transform I 3 into some new matrix, call it D. Gaussian elimination provides a relatively efficient way of constructing the inverse to a matrix. Exactly the same results hold with any number of variables and equations. Gaussian elimination is practical, under most circumstances, for finding the inverse to matrices involving thousands of equations and variables.
The history of Gaussian elimination and its names is quite interesting, you will be surprised to know that the name "Gaussian" was attributed to this methodology by mistake in the last century.
In reality the algorithm to simultaneously solve a system of linear equations using matrices and row reduction has been found to be written in some form in ancient Chinese texts that date to even before our era.
Then in the late 's Isaac Newton put together a lesson on it to fill up something he considered as a void in algebra books. After the name "Gaussian" had been already established in the 's, the Gaussian-Jordan term was adopted when geodesist W. Jordan improved the technique so he could use such calculations to process his observed land surveying data.
If you would like to continue reading about the fascinating history about the mathematicians of Gaussian elimination do not hesitate to click on the link and read. There is really no physical difference between Gaussian elimination and Gauss Jordan elimination, both processes follow the exact same type of row operations and combinations of them, their difference resides on the results they produce.
Many mathematicians and teachers around the world will refer to Gaussian elimination vs Gauss Jordan elimination as the methods to produce an echelon form matrix vs a method to produce a reduced echelon form matrix, but in reality, they are talking about the two stages of row reduction we explained on the very first section of this lesson forward elimination and back substitution , and so, you just apply row operations until you have simplified the matrix in question.
If you arrive to the echelon form you can usually solve a system of linear equations with it up until here, this is what would be called Gaussian elimination. If you need to continue the simplification of such matrix in order to obtain directly the general solution for the system of equations you are working on, for this case you just continue to row-operate on the matrix until you have simplified it to reduced echelon form this would be what we call the Gauss-Jordan part and which could be considered also as pivoting Gaussian elimination.
We will leave the extensive explanation on row reduction and echelon forms for the next lesson, for now you need to know that, unless you have an identity matrix on the left hand side of the augmented matrix you are solving in which case you don't need to do anything to solve the system of equations related to the matrix , the Gaussian elimination method regular row reduction will always be used to solve a linear system of equations which has been transcribed as a matrix.
As our last section, let us work through some more exercises on Gaussian elimination row reduction so you can acquire more practice on this methodology. Throughout many future lessons in this course for Linear Algebra, you will find that row reduction is one of the most important tools there are when working with matrix equations.
Therefore, make sure you understand all of the steps involved in the solution for the next problems. For this system we know we will obtain an augmented matrix with three rows since the system contains three equations and three columns to the left of the vertical line since there are three different variables.
On this case we will go directly into the row reduction, and so, the first matrix you will see on this process is the one you obtain by transcribing the system of linear equations into an augmented matrix. And so, the final solution to this system of equations looks as follows:. We substitute this in the equations resulting from the second and first row in that order to calculate the values of the variables x and y:.
And the final solution to this system of equations is:. To finalize our lesson for today we have a link recommendation to complement your studies: Gaussian elimination an article which contains some extra information about row reduction, including an introduction to the topic and some more examples.
As we mentioned before, be ready to keep on using row reduction for almost the whole rest of this course in Linear Algebra, so, we see you in the next lesson! Solving a linear system with matrices using Gaussian elimination.
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Home Algebra Matrices. Still Confused? Nope, got it. Active 1 year ago. Viewed 9k times. Where can I read the proof?
Add a comment. Active Oldest Votes. That's likely to be confusing to most people asking this question. Spencer Spencer This is probably that what you want to say below the equations. That is the thing that, in my opinion, one needs to spell out. It is the pivot in the sense of Leron of the reasoning. The link I provided goes into significant detail. Thank you for responding. Matrices A and B are in reduced-row echelon form, but matrices C and D are not. C is not in reduced-row echelon form because it violates conditions two and three.
D is not in reduced-row echelon form because it violates condition four.
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