# Invertibility of Matrix with Number Theory

I'm studying for an exam on abstract algebra, and I found a activity which asked you to discuss if the following matrix is invertible: $$\begin{equation} \begin{pmatrix} 54401 & 65432 & 45530 & 45678\\ 34567 & 12121 & 11111 & 12345\\ 12345 & 76543 & 98760 & 65456\\ 43211 & 45678 & 88888 & 98765 \end{pmatrix} \end{equation}$$

I can detect certain partterns in the matrix. However, this question was included in the section of the Integer's Ring and as a hint the activity proposes to do a reduction modulo certain integer. First of all, I can't see a direct relation between singular matrices and $$\mathbb Z/(n)$$ (maybe for finding linear combinations(?), but I'm not sure). I would thank any hint or possible approach to the activity.

• If the determinant is non-zero modulo some integer, the determinant must itself be non-zero, and hence be invertible. Jan 21 at 9:15

Hint: Consider $$A$$ and $$\det(A)$$ modulo $$2$$. It is easy to see that $$\det(A)\equiv 1 \bmod 2$$. In particular, $$\det(A)\neq 0$$.