Finiteness and order of the cokernel of a square integer matrix

If $n=0$, the codomain and image are both the zero group, so the cokernel is trivial and has order $1=\mathrm{\mid }\det A\mathrm{\mid }$. Now suppose $n>0$. Multiplication on the left or right by an integer matrix with determinant $\pm 1$ preserves the isomorphism type of the cokernel and preserves the absolute determinant: left multiplication changes the basis of the target, and right multiplication changes the basis of the domain.

Whenever the current remaining square submatrix is nonzero, among all matrices obtained from it by such row and column operations choose one whose least nonzero absolute entry is as small as possible; by swaps and a sign change, place that positive entry $d$ in position $(1,1)$. If any entry in its first row or column is not divisible by $d$, Euclidean division and a row or column subtraction produce a nonzero remainder strictly between $0$ and $d$, which contradicts the minimality of $d$. Clear the first row and column by subtracting multiples of the first column and row. If an entry $b$ in the remaining submatrix were not divisible by $d$, add its row to the first row, write $b=kd+r$ with $0<r<d$, and subtract $k$ times the first column from its column; swapping that column with the first would again give a smaller nonzero pivot. Hence $d$ divides the remaining submatrix.

Apply the same finite pivot procedure to each remaining square submatrix. If a remaining submatrix is zero, stop and retain it unchanged as part of the zero block. Since every nonzero pivot removes one row and one column, induction gives a diagonal matrix $\mathrm{diag}(d_1,\dots ,d_r,0,\dots ,0)$ with each $d_i>0$, obtained by unimodular row and column operations. Its cokernel is ${⨁}_{i=1}^r\mathbb{Z}\mathrm{/}{d}_i\mathbb{Z}\oplus {\mathbb{Z}}^{n-r}$. Thus it is finite exactly when $r=n$, equivalently when $\det A\mathrm{\ne }0$; then its order is ${\prod }_i{d}_i=\mathrm{\mid }\det A\mathrm{\mid }$. The operations in step 1.1 preserve both conclusions for the original matrix.