Gauss Elimination Method Solved Problems

Gauss Elimination Method Solved Problems-36
So what's the augmented matrix for this system of equations? So the first thing, I have a leading 1 here that's a pivot entry. So let me replace the third row with the third row minus 2 times the second row. 3 minus 2 times 2, that's 3 minus 4, or minus 1.

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For example, in the following sequence of row operations (where multiple elementary operations might be done at each step), the third and fourth matrices are the ones in row echelon form, and the final matrix is the unique reduced row echelon form.

Using row operations to convert a matrix into reduced row echelon form is sometimes called Gauss–Jordan elimination.

It's to the right of this one, which is what I want for reduced row echelon form.

And to zero this guy out, what I can do is I can replace the first row with the first row minus the second row.

My second row is 0, 1, 2, and then I have a minus 3, the augmented part of it.

I'm replacing the first row with the first row minus the second row. The second part (sometimes called back substitution) continues to use row operations until the solution is found; in other words, it puts the matrix into reduced row echelon form.Another point of view, which turns out to be very useful to analyze the algorithm, is that row reduction produces a matrix decomposition of the original matrix.To perform row reduction on a matrix, one uses a sequence of elementary row operations to modify the matrix until the lower left-hand corner of the matrix is filled with zeros, as much as possible.There are three types of elementary row operations: Using these operations, a matrix can always be transformed into an upper triangular matrix, and in fact one that is in row echelon form.The process of row reduction makes use of elementary row operations, and can be divided into two parts.The first part (sometimes called forward elimination) reduces a given system to row echelon form, from which one can tell whether there are no solutions, a unique solution, or infinitely many solutions.I figure it never hurts getting as much practice as possible solving systems of linear equations, so let's solve this one. What I'm going to do is I'm going to solve it using an augmented matrix, and I'm going to put it in reduced row echelon form. Once all of the leading coefficients (the leftmost nonzero entry in each row) are 1, and every column containing a leading coefficient has zeros elsewhere, the matrix is said to be in reduced row echelon form.This final form is unique; in other words, it is independent of the sequence of row operations used.


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