Theorem: Let A be nxn matrix on real numbers, then<\/p>\n\n\n\n
Det(A\u1d57)=Det(A)<\/p>\n\n\n\n
Proof: Let e be any elementary row operation on nxn matrices. Then, define e’ to be analogous column operation on nxn matrices. What do we mean by the word “analogous” here? Well, let’s say e multiplies r’th row by c and adds it to s’th row. Then, e’ multiplies r’th column by c and adds it to s’th column. If e multiplies r’th row by c that isn’t zero, then e’ multiplies r’th row by c. Then, there are some interesting consequences from this definition.<\/p>\n\n\n\n
Proposition-1: Let A be an nxn matrix on real numbers. Also, let e be any elementary row operation. Finally, define e’ to be analogous column operation as defined above. Then,<\/p>\n\n\n\n
(e(A))t<\/sup>=e'(A\u1d57)<\/p>\n\n\n\n Proof: <\/p>\n\n\n\n ero-1: Let<\/p>\n\n\n\n (e(A))(i,j)<\/sub>={ A(i,j)<\/sub> if i\u2260r ; c*A(s,j)<\/sub>+A(i,j)<\/sub> if i=r}<\/p>\n\n\n\n then <\/p>\n\n\n\n (e(A))t<\/sup>(j,i)<\/sub>={ A(i,j)<\/sub> if i\u2260r ; c*A(s,j)<\/sub>+A(i,j)<\/sub> if i=r}<\/p>\n\n\n\n also<\/p>\n\n\n\n (A)t<\/sup>(j,i)<\/sub>={A(i,j)<\/sub>}<\/p>\n\n\n\n (e'(A)t<\/sup>)(j,i)<\/sub>={At<\/sup>(j,i)<\/sub> if i\u2260r ; c*At<\/sup>(j,s)<\/sub>+At<\/sup>(j,i)<\/sub> if i=r}<\/p>\n\n\n\n this equality becomes,<\/p>\n\n\n\n (e'(A)t<\/sup>)(j,i)<\/sub>={A(i,j)<\/sub> if i\u2260r ; c*A(s,j)<\/sub>+A(i,j)<\/sub> if i=r}<\/p>\n\n\n\n ero-2: Let<\/p>\n\n\n\n (e(A))(i,j)<\/sub>={ A(i,j)<\/sub> if i\u2260r ; c*A(i,j)<\/sub> if i=r and c\u22600}<\/p>\n\n\n\n then <\/p>\n\n\n\n (e(A))t<\/sup>(j,i)<\/sub>={ A(i,j)<\/sub> if i\u2260r ; c*A(i,j)<\/sub> if i=r and c\u22600}<\/p>\n\n\n\n also<\/p>\n\n\n\n (A)t<\/sup>(j,i)<\/sub>={A(i,j)<\/sub>}<\/p>\n\n\n\n (e'(A)t<\/sup>)(j,i)<\/sub>={At<\/sup>(j,i)<\/sub> if i\u2260r ; c*At<\/sup>(j,i)<\/sub> if i=r and c\u22600}<\/p>\n\n\n\n this equality becomes,<\/p>\n\n\n\n (e'(A)t<\/sup>)(j,i)<\/sub>={A(i,j)<\/sub> if i\u2260r ; c*A(i,j)<\/sub> if i=r and c\u22600 }<\/p>\n\n\n\n ero-3: Let<\/p>\n\n\n\n (e(A))(i,j)<\/sub>={ A(i,j)<\/sub> if i\u2260r,s ; A(s,j)<\/sub> if i=r ; A(r,j)<\/sub> if i=s}<\/p>\n\n\n\n then <\/p>\n\n\n\n (e(A))t<\/sup>(j,i)<\/sub>={ A(i,j)<\/sub> if i\u2260r,s ; A(s,j)<\/sub> if i=r ; A(r,j)<\/sub> if i=s}<\/p>\n\n\n\n also<\/p>\n\n\n\n (A)t<\/sup>(j,i)<\/sub>={A(i,j)<\/sub>}<\/p>\n\n\n\n (e'(A)t<\/sup>)(j,i)<\/sub>={At<\/sup>(j,i)<\/sub> if i\u2260r ; At<\/sup>(s,i)<\/sub> if i=r ; At<\/sup>(r,i)<\/sub> if i=s}<\/p>\n\n\n\n this equality becomes,<\/p>\n\n\n\n (e'(A)t<\/sup>)(j,i)<\/sub>={A(i,j)<\/sub> if i\u2260r ; A(i,s)<\/sub> if i=r ; A(i,r)<\/sub> if i=s}<\/p>\n\n\n\n So, proposition is proved.<\/p>\n\n\n\n Proposition-2: Let A be any nxn matrix on real numbers. Let e be any elementary row operation. Also, let e’ be analogous column operation. Then,<\/p>\n\n\n\n e'(A)=A.(e'(I))<\/p>\n\n\n\n Proof: Let A<\/p>\n\n\n\n <\/p>\n\n\n\n <\/p>\n\n\n\n <\/p>\n\n\n\n <\/p>\n\n\n\n <\/p>\n","protected":false},"excerpt":{"rendered":" Theorem: Let A be nxn matrix on real numbers, then Det(A\u1d57)=Det(A) Proof: Let e be any elementary row operation on nxn matrices. Then, define e’ to be analogous column operation on nxn matrices. What do we mean by the word “analogous” here? Well, let’s say e multiplies r’th row by c and adds it to … <\/p>\n