How Does Subtracting c from ann Affect Matrix Singularity?

[Ylle]

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_1nA_1n+...+a_nnA_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_1nA_1n+...+a_nnA_nn

Then if I subtract cA_nn from both sides i get:

0 = a_1nA_1n+...+a_nnA_nn - cA_nn
, which we can rewrite to:

0 = a_1nA_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 ? :)

 

[jbunniii]

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_1nA_1n+...+a_nnA_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_1nA_1n+...+a_nnA_nn

Then if I subtract cA_nn from both sides i get:

0 = a_1nA_1n+...+a_nnA_nn - cA_nn
, which we can rewrite to:

0 = a_1nA_1n+...+(a_nn-c)A_nnAnd 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 ? :)

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.