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Matrix singular problem

  • Thread starter Ylle
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  • #1
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Homework Statement


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.


Homework Equations



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 ? :)
 

Answers and Replies

  • #2
jbunniii
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Homework Statement


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.


Homework Equations



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

[tex]a_{nn} - c[/tex] instead of [tex]a_{nn}[/tex]

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

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