Gauss-Jordan elimination means you find the matrix inverse A −1. Non-invertible matrices (also known as singular matrices) are not as helpful when trying to solve a system exactly. If you row-reduce a matrix and it does not become an identity matrix in RREF, then that matrix was non-invertible. The most useful property for your question is that their RREF is the identity matrix (a matrix with only 1’s down the diagonal and 0’s everywhere else). Invertible matrices have several important properties. In a system Ax=B, x can only be solved for if A is invertible. See these Khan Academy videos for worked examples. If you go through each pivot (the numbers along the diagonal) and divide those rows by their leading coefficient, then you will end up in RREF.
My interpretation of REF is just doing row operations in such a way to avoid dividing rows by their pivot values (to make the pivot become 1). Such row operations usually involve multiplying/dividing by nonzero scalar multiples of the row, or adding/subtracting nonzero scalar multiples of one row from another row. For this to happen, you must perform row operations to “pivot” along each entry along the diagonal. In your question, you say you reduce a matrix A to a diagonal matrix where every nonzero value equals 1.
Also, Gaussian elimination is all you need to determine the rank of a matrix (an important property of each matrix) while going through the trouble to put a matrix in reduced row echelon form is not worth it to only solve for the matrix’s rank.ĮDIT: Here are some abbreviations to start off with: REF = “Row Echelon Form”. However, Gaussian elimination in itself is occasionally computationally more efficient for computers. For small systems (or by hand), it is usually more convenient to use Gauss-Jordan elimination and explicitly solve for each variable represented in the matrix system. Gaussian Elimination helps to put a matrix in row echelon form, while Gauss-Jordan Elimination puts a matrix in reduced row echelon form. This guy (the The Organic Chemistry Tutor) explains it so well…my textbook explains it like a brick. My professor makes this seem so complicated but when you (the The Organic Chemistry Tutor) didn’t it was so easy. My math teacher totally confused the class with this principle. Hey, it’s 2021 and these videos from 13 years ago ( The Khan Academy) are saving my university economics degree, thank you This method is a step‐by‐step elimination of the variables. Once this final variable is determined, its value is substituted back into the other equations in order to evaluate the remaining unknowns. The fundamental idea of Gaussian elimination is to add multiples of one equation to the others in order to eliminate a variable and to continue this process until only one variable is left. Gauss Elimination & Gauss – Jordan Elimination and also differences between both – Edu Society Definition
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