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## Homework Statement

Let

*A*be a non-singular n x n matrix with a non-zero cofactor

*A*

_{nn}and let

*c*= det(A) /

*A*

_{nn}

Show that if we subtract

*c*from

*a*

_{nn}, then the resulting matrix will be singular.

## Homework Equations

det(A) =

*a*

_{1n}

*A*

_{1n}+...+

*a*

_{nn}

*A*

_{nn}

## The Attempt at a Solution

Well, if I replace det(A) with the one in "Relevant eq.", and multiply both sides with

*A*

_{nn}I get:

*cA*

_{nn}=

*a*

_{1n}

*A*

_{1n}+...+

*a*

_{nn}

*A*

_{nn}

Then if I subtract

*cA*

_{nn}from both sides i get:

0 =

*a*

_{1n}

*A*

_{1n}+...+

*a*

_{nn}

*A*

_{nn}-

*cA*

_{nn}

, which we can rewrite to:

0 =

*a*

_{1n}

*A*

_{1n}+...+(

*a*

_{nn}-c)

*A*

_{nn}

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 ? :)