Let A be a non-singular n x n matrix with a non-zero cofactor Ann and let
c = det(A) / Ann
Show that if we subtract c from ann, then the resulting matrix will be singular.
det(A) = a1nA1n+...+annAnn
The Attempt at a Solution
Well, if I replace det(A) with the one in "Relevant eq.", and multiply both sides with Ann I get:
cAnn = a1nA1n+...+annAnn
Then if I subtract cAnn from both sides i get:
0 = a1nA1n+...+annAnn - cAnn
, which we can rewrite to:
0 = a1nA1n+...+(ann-c)Ann
And now I'm not sure if I'm done ?
It seems like I need to define another matrix of some sort to define det(B) = 0.
But I'm not quite sure how I do that. Can anyone give me a hint ? :)