Chapter 3 - Gauss-Jordan Elimination and Reduced Row Echelon Form

Section 3.1 - Gauss-Jordan Elimination and rref

Example 3.1.1

CM={{6,1,0,4,-3},{-9,2,3,-8,1},{7,0,-4,5,2}};

RowReduce[CM]//MatrixForm

RowReduce[{CM[[3]],CM[[2]],CM[[1]]}]//MatrixForm

RowReduce[{CM[[1]],CM[[3]],CM[[2]]}]//MatrixForm

Solve[{6 x+y+4 w==-3,-9 x+2 y+3 z-8 w==1,7 x-4 z+5 w==2},{x,y,z}]

Solve[{6 x+y+4 w==-3,-9 x+2 y+3 z-8 w==1,7 x-4 z+5 w==2},{y,z,w}]

Example 3.1.2

CM={{6,1,0,4,-3},{-9,2,3,-8,1},{7,0,-1,5,2}};

RowReduce[CM]//MatrixForm

Solve[{6 x+y+4 w==-3,-9 x+2 y+3 z-8 w==1,7 x-z+5 w==2},{x,y,w}]

Solve[{6 x+y+4 w==-3,-9 x+2 y+3 z-8 w==1,7 x-z+5 w==2},{x,y,z}]

Example 3.1.3

CM={{6,1,0,4,-3},{-9,2,3,-8,1},{-3,3,3,-4,10}};

RowReduce[CM]//MatrixForm

Solve[{6 x+y+4 w==-3,-9 x+2 y+3 z-8 w==1,-3 x+3 y+3 z-4 w==10},{x,y,z}]

Example 3.1.4

CM={{6,1,0,4,-3},{-9,2,3,-8,1},{-3,3,3,-4,-2}};

RowReduce[CM]//MatrixForm

Solve[{6 x+y+4 w==-3,-9 x+2 y+3 z-8 w==1,-3 x+3 y+3 z-4 w==-2},{x,y}]

Example 3.1.5

CM={{-7,2,-9,3,-5},{4,1,6,-11,15},{1,-1,3,8,4},{9,12,-7,5,-2}};

(R=RowReduce[CM])//MatrixForm

R[[All,-1]]

Example 3.1.6

CM={{7,2,9,8},{-3,5,6,-2},{11,-1,4,3},{8,13,-10,-11}};

RowReduce[CM]//MatrixForm

CM2={{7,2,9,8},{-3,5,6,-2},{11,-1,4,3}};

RowReduce[CM2]//MatrixForm

(CM3={{7,2,9,8},{-3,5,6,-2}})//MatrixForm

RowReduce[CM3]//MatrixForm