Saturday, 8 March 2014

linear algebra - Calculating the determinant as a product without making any calculations



My problem is on the specific determinant.




det(na1+b1na2+b2na3+b3nb1+c1nb2+c2nb3+c3nc1+a1nc2+a2nc3+a3)=(n+1)(n2n+1)det(a1a2a3b1b2b3c1c2c3)



All I can do is prove the factor (n+1) and I think that we have to work only on one column and the do the exact same thing to the others.


Answer



Use multilinearity of determinant:



|na1+b1na2+b2na3+b3nb1+c1nb2+c2nb3+c3nc1+a1nc2+a2nc3+a3|=|na1na2na3nb1nb2nb3nc1nc2nc3|+|na1na2b3nb1nb2c3nc1nc2a3|+



+|na1b2na3nb1c2nb3nc1a2nc3|+|na1b2b3nb1c2c3nc1a2a3|+|b1na2na3c1nb2nb3a1nc2nc3|+



Observe that if we put



Δ=|a1a2a3b1b2b3c1c2c3|




then we have that the four first determinants above equal (factor out constants from rows/columns):



n3Δ+n2Δ+n2|a1b2a3b1c2b3c1a2c3|+n|a1b2b3b1c2c3c1a2a3|+



Well, develop the other three determinants left and sum up all.


No comments:

Post a Comment

real analysis - How to find limhrightarrow0fracsin(ha)h

How to find limh0sin(ha)h without lhopital rule? I know when I use lhopital I easy get $$ \lim_{h\rightarrow 0}...