Chapter 5 - Determinants, Inverses, and Cramer’s Rule

Section 5.2 - Determinants by Expanding Along Any Row or Column

Example 5.2.1

(A={{-9,5,0,2},{4,-1,3,7},{6,-2,0,-8},{1,10,-4,11}})//MatrixForm

Drop[A,{2},{3}]//MatrixForm

Det[%]

Sum[(((-1))^(i+3))A[[i,3]] Det[Drop[A,{i},{3}]],{i,1,4}]

Sum[(((-1))^(1+j))A[[1,j]] Det[Drop[A,{1},{j}]],{j,1,4}]

Det[A]

Sum[(((-1))^(3+j))A[[2,j]] Det[Drop[A,{3},{j}]],{j,1,4}]

Sum[(((-1))^(i+3))A[[i,1]] Det[Drop[A,{i},{3}]],{i,1,4}]

Example 5.2.2

A={{a,b,c},{d,e,f},{g,h,k}};

Sum[(((-1))^(i+1))A[[i,1]] Det[Drop[A,{i},{1}]],{i,1,3}]

Simplify[%]

Det[A]

Sum[(((-1))^(3+j))A[[3,j]] Det[Drop[A,{3},{j}]],{j,1,3}]

Simplify[%]

Commands after Example 5.2.3

A={{a,b},{c,d}};

B={{e,f},{g,h}};

Det[A.B]

Expand[Det[A] Det[B]]