Monday, 1 August 2016

linear algebra - Intuition behind Matrix being invertible iff determinant is non-zero

I have been wondering about this question since I was in school. How can one number tell so much about the whole matrix being invertible or not?



I know the proof of this statement now. But I would like to know the intuition behind this result and why this result is actually true.






The proof I have in mind is:



If A is invertible, then




1=det



whence \det(A) \neq 0.



Conversely, if \det(A) \neq 0, we have



A adj(A) = adj(A)A = \det(A)I



whence A is invertible.




adj(A) is the adjugate matrix of A.



adj(A)_{ji} = (-1)^{i+j}\det(A_{ij})



where A_{ij} is the matrix obtained from A by deleting ith row and jth column.



Any other insightful proofs are also welcome.

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