# Matrix singular problem

1. May 13, 2009

### Ylle

1. The problem statement, all variables and given/known data
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.

2. Relevant equations

det(A) = a1nA1n+...+annAnn

3. 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 ? :)

2. May 13, 2009

### jbunniii

Yes, define a matrix B which is exactly the same as A except for the (n,n) element, which will be

$$a_{nn} - c$$ instead of $$a_{nn}$$

Then calculate det(B) the same way you calculated det(A), and compare the answers you get.