Show a matrix is invertible

Question

How to show that $$A=\begin{pmatrix}1233&2344&1324&3456\\ 2342&11233&1432&13256\\234132&32432&1234567&43254\\423412&42354&452356&13245\end{pmatrix}$$ is invertible ? I could compute the determinant, but it's very very long...

@SamiBen: Oh, true, I had already seen this link long times ago, but I had totally forgot the way to solve it. Thanks. — idm, Feb 25 '15 at 12:15

score 14 · Answer 1 · answered Feb 25 '15 at 11:59

14

If you take the coefficients modulo 2, you see that there is only $1$ on the diagonal and $0$ elsewhere. Therefore $$\det A\equiv 1\pmod 2,$$ and thus $\det A$ is odd. Therefore $\det A\neq 0$ and thus $A$ is invertible.

answered Feb 25 '15 at 11:59

Surb

55,662

Very nice trick! – yurnero Feb 25 '15 at 12:03
sweet. I would give 10 upvotes if I could. – 5xum Feb 25 '15 at 12:06
@Surb: Thanks for the trick :-) – idm Feb 25 '15 at 12:16

Show a matrix is invertible

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