Row Reduction

Row Reduction

This is an alternate method of finding the Inverse Matrix; Any 2 rows can be interchanged Any row/column can be multiplied by a non-0 constant Any row/column can be replaced by itself, plus any other row/column multiplied by a non-zero constant Any row/column can be replaced by itself multiplied by a non-0 constant, plus any other row/column multiplied by a different non-0 constant NB: This is called a Row Reduced Matrix

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Echelon Form

The number of 0's preceding the 1st non-0 entry increases row by row.NB: Try to make as many 0 terms as possible.

Echelon Form (image/png)

Caption: : The best Echelon Matrix

Solutions of Equations

After expressing in the correct form, use Row Reduction methods on both sides to change the matrix into Echelon Form.For a 3x3 matrix, try to get either Row 2 or Row 3 to have 2 0's as this will make it easier to find either y or z.Then substitute this answer into the other equations to find x and either y or z (whichever one you didn't find 1st).

Solutions Of Equ (image/png)

Equations with 2 Unknowns

Two LinesConsider 2 straight lines ax+by=m and cx+dy=n then; If the lines are parallel, then there are no solutions and the equations are inconsistant 2 lines will intersect at one unique point If the 2 lines are the same, there are an infinate number of solutions

Three Linesax+by+m=0cx+dy+n=0ex+fy+p=0These are consistant if the determinant = 0

Equations with 3 Unknowns

ax+by+cz=m is the equation of a planeax+by+cz=m and ax+by+cz=n where m≠n represent parallel planesIf 2 equations are multiples of each other, then they represent the same planeNB: Solving a set of 3 equations with 3 unknowns is finding where the 3 planes, represented by the 3 equations, intersect.