Determinants

A determinant is the are/volume formed by a set of vectors.

b={1,1.5};
MatrixForm[b];
orig={0,0};
l[b_,x_,y_]:=Line[{orig,b,b+{x,y},{x,y},orig}];
det[x_,y_]:=Det[b,{x,y}];

pic1[x_,y_]:=Graphics[{l[b,x,y],Thick,Red,Arrow[{orig,b}],Thick,Green,Arrow[{orig,{x,y}}]},AspectRatio->1,Axes->True,PlotRange->{{-3,3},{-3,3}}];
pic2[x_,y_]:=Graphics[{Green,Polygon[{{0,0},{0.3,0},{0.3,Det[{b,{x,y}}]},{0,Det[{b,{x,y}}]},{0,0}},VertexColors->{Green,Green, Black,Black}]},Axes->{False,True},AspectRatio->6,PlotRange->{-4,4}];

Manipulate[{pic1[x,y],pic2[x,y]},
{x,2,-2,-.1},{y,2,-2,-.1}]

3 D vectors create volumes.

v1={1,2,3};
v2={-1,-1,0};
v3={2,0,3};
Det[{v1,v2,v3}];

Interchange two vectors : Det changes sign.

Det[{v1,v3,v2}];

Multiply a vector by a constant : Det gets multiplied by that constant.

Det[{2v1,v2,v3}];

Linearly dependent vectors : Det is zero.

Det[{v1,v2,v1+2v2}];

A diagonal matrix

MatrixForm[{{r,0,0},{0,s,0},{0,0,t}}];
Det[{{r,0,0},{0,s,0},{0,0,t}}];

A 4 D example

w1=Table[RandomReal[],{i,1,4}];w2=Table[RandomReal[],{i,1,4}];w3=Table[RandomReal[],{i,1,4}];w4=Table[RandomReal[],{i,1,4}];

mat={w1,w2,w3,w4};
MatrixForm[mat];
Det [mat];
Det[{w1,w3,w2,w4}];
MatrixRank[mat];

A 10 D example

mat10=Table[i*j,{i,1,10},{j,1,10}];
MatrixForm[mat10];
Det[mat10];
MatrixRank[mat10];