Who established the tradition of using $X^{\prime}$ instead of $X^{T}$ to denote the matrix transpose?

Question

From being away from mathematicians for a while and spending most of my time with econometricians and statisticians, one thing I've noticed is that econometricians and statisticians like to use $\prime$ to denote the matrix transpose, e.g., $X^{\prime}$.

However, when you show this notation to a mathematician, they'd think that you mean a matrix distinct from $X$; i.e., $X$ and $X^{\prime}$ are just two distinct matrices with no explicit relation. Hence when I'm on MSE, I always try to use $X^{T}$ instead.

My guess is that the $X^{T}$ notation has been around longer than $X^{\prime}$. Who was the first to use $X^{\prime}$ to denote the matrix transpose?

Answer 1

In A History of Mathematical Notations (vol. 2) by Florian Cajori (1929) on p. 103, the use of $A^{\prime}$ to denote the "conjugate" of a matrix $A$ (what we now usually call the "transpose") is credited to

C.E. Cullis, Matrices and Determinoids (Cambridge), Vol. I (1913)

This notation is defined in p. 5 of that text. From kimchi lover's answer, it appears that $A^{\prime}$ predates $A^{T}$.

Answer 2

I did a spot check on some books:

Prime
Birkhoff and MacLane, Survey of Modern Algebra (1953), Faddeeva, Computational Methods of Linear Algebra (trans 1959), and P. Halmos, Finite Dimensional Vector Spaces (1958) all use $A'$ for the transpose of $A$. As do Wedderburn Lectures on Matrices (1934) and Turnbull and Aitken Theory of Canonical Matrices (1932).

T
But Macduffee Vectors and Matrices (1942) and Gantmacher Matrix Theory (1953, trans 1959) both use $A^T$. Macduffee's 1933 The Theory of Matrices has some relevant info: on page 5, when the term transpose and his symbol $A^T$ first appear, he gives a footnote reading

Or conjugate. Many different notations for the transpose have been used, as $A'$, $\bar A$, $\breve{A}$, $A^*$, $A_1$, ${}_tA$. The present notation is in keeping with a systematic notation which, it is hoped, may find favor.

This makes me think Macduffee introduced the $A^T$ notation, and that the $A'$ notation predates it.

Neither
The local library has Bocher, Introduction to Higher Algebra (1907) which does not have a special symbol for transpose.