# Non-zero solution of a homogenous system

## Main Question or Discussion Point

Hi,
Could anyone give me a proof for the following theorem?
Theorem : Ax=0 has a non-trivial solution iff det(A)=0

/Sunny

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You will find the proof in books such as "Linear Algebra" by Lang. I suggest you check out that book.

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This is easier to understand if you think about the geometric meaning of a determinant.

A matrix $A$ represents a linear transformation in a given basis. The determinant is a property of the matrix, but it's also a property of the linear transformation it represents, independently of any basis. The determinant is simply the signed volume change of the volume element.

Now we can see what a zero determinant means: it means the transformed volume element is zero. This transformation has squished it down into a lower-dimensional subspace, which means the mapped basis vectors can't be linearly independent. That means there's a nontrivial linear combination of basis vectors which maps to 0. In other words, there must be some $x$ in this basis such that $Ax = 0$.

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