2016.01.25 Linear Algebra Class

Elementary Matrices + Row Reduction

Solve[{3x+5y-7z==1, 2x-2y+z==2,5x+15y+3z==4},{x,y,z}]

A = {{3,5,-7,1},{2,-2,1,2},{5,15,3,4}};

A//MatrixForm

RowReduce[A]//MatrixForm

E1 = {{1/3,0,0},{0,1,0},{0,0,1}};

E1//MatrixForm

E1.A//MatrixForm

E2 = {{1,0,0},{-2,1,0},{0,0,1}};

E2//MatrixForm

E2.E1.A//MatrixForm

E3 = {{1,0,0},{0,1,0},{-5,0,1}};

E3//MatrixForm

E3.E2.E1.A//MatrixForm

E4 = {{1,0,0},{0,-3/16,0},{0,0,1}};

E4.E3.E2.E1.A//MatrixForm

E5 = {{1,-5/3,0},{0,1,0},{0,0,1}};

E5//MatrixForm

E5.E4.E3.E2.E1.A//MatrixForm

E6 = {{1,0,0},{0,1,0},{0,-20/3,1}};

E6//MatrixForm

E6.E5.E4.E3.E2.E1.A//MatrixForm

E7 = {{1,0,0},{0,1,0},{0,0,4/87}};

E7.E6.E5.E4.E3.E2.E1.A//MatrixForm

E8 = {{1,0,9/16},{0,1,0},{0,0,1}};

E8//MatrixForm

E8.E7.E6.E5.E4.E3.E2.E1.A//MatrixForm

E9 = {{1,0,0},{0,1,17/16},{0,0,1}};

E9.E8.E7.E6.E5.E4.E3.E2.E1.A//MatrixForm

E9.E8.E7.E6.E5.E4.E3.E2.E1//MatrixForm

B = {{3,5,-7,0},{2,-2,1,-2},{5,15,3,3}};

B//MatrixForm
A//MatrixForm

E9.E8.E7.E6.E5.E4.E3.E2.E1.B//MatrixForm

A2 = {{3,5,-7,1},{2,-2,1,2}};

A2//MatrixForm

F1 = {{1/3,0},{0,1}};

F1.A2//MatrixForm

F2 = {{1,0},{-2,1}};

F2.F1.A2//MatrixForm

F3 = {{1,0},{0,-3/16}};

F3.F2.F1.A2//MatrixForm

F4 = {{1,-5/3},{0,1}};

F4.F3.F2.F1.A2//MatrixForm

F4.F3.F2.F1//MatrixForm

A//MatrixForm

Eswap = {{0,1,0},{1,0,0},{0,0,1}};

Eswap//MatrixForm

Eswap.A//MatrixForm