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Let $A$ a square matrix with the size of $n \times n$.

I know that if the rank of the matrix is $<n$, then there must be a "zeroes-line", therefore $\det(A)=0$.

What about $\text{rank}(A)=n$? Why does it imply $\det(A)\ne0$?

Of course, there is no "zeroes-line", but that doesn't prove it yet.
I've seen a proof in a book which does this conclusion immediately, but IMHO this alone, doesn't prove it.

What's the missing part?

user26857
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AndrePoole
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3 Answers3

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Let $A$ be an $n\times n$ matrix.

Note that $\det(A) \neq 0$ iff the rows are linearly independent iff $rank(A)=n$.

tonytonov
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Kuai
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The rank of $A$ can be viewed as $m$ where $m$ is the size of the largest non-zero $m\times m$ submatrix with non-zero determinant.

Alternatively, you can row reduce the matrix to give you an upper triangular matrix using row interchanges and adding scalar multiples of a row to another row. This will only affect the sign of the determinant. If an $n \times n$ matrix has rank $n$ then it has $n$ pivot columns (and therefore $n$ pivot rows). This means you will be able to row reduce it to an upper triangular form with pivots along the diagonal. The determinant is the product of these elements along the diagonal. Can you prove that? Pivots are necessarily non-zero and therefore their product is non-zero, regardless of sign.

CPM
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  • nice, I think you can go all the way and reduce it to the identity matrix as well. and then, $det(I)\ne0$ – AndrePoole Jan 06 '14 at 10:09
  • You sir are correct. This in fact characterizes all invertible $n \times n$ matrices. You will see that it is part of the invertible matrix theorem. – CPM Jan 06 '14 at 10:13
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    @CPM "The rank of $A$ can be viewed as $m$ where $m$ is the size of the largest non-zero $m×m$ submatrix with non-zero determinant." Do you know of a rigorous proof of this statement using elemental methods? – dazedviper Jan 06 '14 at 12:22
  • It is often taken as the definition of rank of a matrix. I see a proof of the "determinant rank" being the same as the "row rank" in the book Elementary Linear Algebra by Kenneth Kuttler, which I see in google books. They come as Theorem 8.5.7 and Corollary 8.5.8. The proof looks pretty elementary to me. – CPM Jan 06 '14 at 12:30
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I have an idea: if $$rank(A_{nxn}) = n $$ then $$A$$ must be invertible (proof : https://sharmaeklavya2.github.io/theoremdep/nodes/linear-algebra/matrices/full-rank-inv.html) so $$A^{-1} = \frac{adj(A)}{det(A)}$$ exists and therefore $$\rightarrow det(A) \neq 0$$

tonythestark
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