Reduced row echelon form | Lecture 11 | Matrix Algebra for Engineers

we've talked about Gaussian elimination where we bring a matrix to upper triangular form there's a theoretical construct called the reduced row echelon form has this somewhat complicated name where we go even further than upper triangular matrix we basically go all the way to possibly the identity matrix here we won't go to the identity matrix but the idea is that you use these pivots to eliminate not just below the pivot but also above the pivot so I think the easiest way to see what the reduced row echelon form of a matrix is is to work an example so I take this rather funny looking matrix matrix as an example one two three four four five six seven six seven eight nine and then let's bring this matrix then to reduce row echelon form and see what it looks like so we start with pivot so that's the pivot again we and we want to eliminate all of the numbers below the pivot so that means we multiply the first row by minus four and we add it to the second row maybe it's helpful if I put the multiplier next to the row so you can see what we're doing so let's go so we got one two three four in the first row and then we multiplied by minus four and add it to the second row so we got our zero here then minus eight plus five is -3 minus 12 plus 6 is minus 6 and minus 16 plus 7 right minus 16 plus 7 is minus 9 I have to practice my arithmetic here and then we want to eliminate the six so in this case we're going to multiply by minus six so let me just write the minus six here so you can remember so then we get a zero right that's the point zero so minus 12 plus 7 is minus 5 and minus 18 plus 8 is minus 10 right and then the last one will be minus 24 plus 9 will be minus 15 ok now we have our new pivot that's our minus 3 here but remember what can we do with this matrix we can switch rows which we don't need to do here but we can also multiply rows by constants so you see the second row has a common factor of 3 and the third row has a common factor of 5 so we might as well make our algebra easier and just divide the second row by minus 3 and the third row by minus 5 so let's do that then the first row is still 1 2 3 4 the second row then becomes 0 1 and then we're dividing through by minus 3 so we have 2 3 here ok and then the last row we divide through by minus 5 so we have 0 1 2 3 ok so that will make our algebra a bit easier so we have our pivot I pivot here is a 1 and we multiply the second row by minus 2 here there's the multiplier and add it to the first row so we can eliminate the 2 so that will go to [Music] one-zero so here we're eliminating both above the pivot and below the pivot so that's the difference with Gaussian elimination where we only eliminate below the pivot so here minus two so that becomes zero minus four plus three is minus 1 and minus six plus four is minus 2 the second row is 0 1 2 3 and the third row well the third row is the same as the second row so that's rather unusual we can multiply the second row by minus 1 here and add it to the third row so the third world just becomes yeah all right what does it mean that two rows are the same it means that the equations they represent are the same so you can just get rid of one of the equations so that whole row becomes 0 so this is called the reduced row echelon form of the matrix so we we write reduced row echelon form right reduced our row echelon form of our matrix a then becomes this matrix here 1 0 minus 1 minus 2 0 1 2 3 & 0 0 0 0 ok so this will be theoretically this will be an important type of matrix to take a matrix and then bring it to reduced row echelon form the key here is that the pivots these are the pivots pivots are supposed to be all one okay so pivots become one and you end up with zeros below and above every pivot ok that's reduced row-echelon form these two columns that have pivots right only two of the columns have pivots two of the columns do not have any pivots the columns that have pivots what are they called pivot columns okay these are called pivot columns okay so this matrix then the original matrix has only two pivot columns so let me review the reduced row echelon form of a matrix is what you get when you use this process of row elimination similar to Gaussian elimination row elimination to convert this matrix into columns that have pivots where the pivots are 1 and 0 above and below the pivot and columns that don't have pivots okay so pivot columns and non-pivot columns later when we do more theoretical things with vector spaces we'll see why it's important to know given a matrix how many columns our pivot columns how many pivots are non-pivot columns I'm Jeff Chas Knopf thanks for watching and I'll see you in the next video you

Channel: Jeffrey Chasnov

Views: 69,624

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Keywords: linear algebra, matrix algebra, Gaussian elimination, reduced row echelon form, rref, gaussian elimination method

Id: 1rBU0yIyQQ8

Length: 8min 22sec (502 seconds)

Published: Mon Jul 09 2018