From todays Quiz (#12):

A = {{2,0,0},{1,1,0},{-1,3,-3}};
B = {{2,0,0},{0,-1,0},{0,0,3}};
Cm = {{-2,4,2},{0,4,2},{0,0,3}};

A//MatrixForm

B//MatrixForm

Cm//MatrixForm

Transpose[B]//MatrixForm

Det[A]

Det[B]

4^3*(-6)

Det[4 B]

Det[Cm]

(-2)^3*1/Det[A]*Det[B]*Det[Cm]

Det[-2 Inverse[A] Transpose[B] Cm]

Before row reducing to get an inverse, see problem 8 on HW part of Section 3.1 for a nice hint....

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

Solve[{3x+4y==9,4x-2y==-10},{x,y}]

AMatrix = {{3,4},{4,-2}};

Inverse[AMatrix].{{b1},{b2}}//MatrixForm

%/.{b1->-1, b2-> 6}

%%/.{b1->9, b2-> -10}

RowReduce[{{3,4,-1,9},{4,-2,6,-10}}]//MatrixForm

(AAugment = {{3,4,1,0},{4,-2,0,1}})//MatrixForm

RowReduce[AAugment]//MatrixForm

Inverse[AMatrix]//MatrixForm

(AAugment1 = {{3,4,1},{4,-2,0}})//MatrixForm

RowReduce[AAugment1]//MatrixForm

SystemEqns={{1,2,-3,4},{3,2,-3,5},{4,5,-2,0}}

SystemEqns//MatrixForm

RowReduce[SystemEqns]//MatrixForm

{{1,4,-3},{3,5,-3},{4,0,-2}}//MatrixForm

(({{1, 4, -3}, {3, 5, -3}, {4, 0, -2}}))

Det[{{1,4,-3},{3,5,-3},{4,0,-2}}]

Det[{{1,2,-3},{3,2,-3},{4,5,-2}}]

26/(-22)

Det[{{1,2,4},{3,2,5},{4,5,0}}]