- #1

mzh

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## Homework Statement

Show that the equation Ac = 0, where A is a NxN matrix and c is a column matrix with elements c_i, i=1..N can have a non-trivial solution (c != 0) only when det(A) = 0.

## Homework Equations

inv(A) does not exist when det(A) = 0.

## The Attempt at a Solution

If det(A) != 0, we can form inv(A). Then we can write

inv(A)*A*c=inv(A)*0

and so

c=0,

and so we arrive at the trivial solution. On the contraire, if det(A)=0, we can not form the inverse, and so we can not write

inv(A)*A*c=inv(A)*0, because inv(A) does not exist.

So, there must be other solutions.

Does that make sense? How do I write this in a more formal way?

Thank you for any suggestions