Q:

What is $\mid\text{det}(A,G)\mid$?

I am reading an old paper dated back in 70', where I encounter this
$$\mid\text{det}(A,G)\mid=(\text{det}\{(A,G)'(A,G)\})^{\frac{1}{2}}.$$
We compute the determinant of a single matrix, don't we? What doest it mean by $\mid\text{det}(A,G)\mid$?

A:

Based on the dimensions of $A$ and $G$ (as defined in the paper), it appears that $\det(A,G)$ is the determinant of the matrix you get by augmenting $A$ by $G$. The identity you mention would be written as
$$\left|\det(M)\right| = \sqrt{\det (M^\top M)}$$
in modern notation.

