I did the calculations for my Ph.D. thesis at the Computation Centre at the University of Toronto on an IBM 602A Calculating Punch, an electromechanical device used in the 1950s for commercial and scientific work. Input was read from punched cards, the basic arithmetic operations of addition, subtraction, multiplication and division were performed, and the results were punched either into the cards from which they were read or some succeeeding card. The program consisting of up to 12 sequential steps was specified by a plugboard. Ancillary equipment such as a keypunch, sorter, tabulator and collator was required.
An excellent description of the use of the 602A for matrix inversion is given by Bose and Roy. The algorithm for matrix inversion requires a sequence of 17 steps with embedded loops describing the repeated sorting, reproducing and punching of cards on the ancillary equipment. The use of the 602A is given in Step (9): "Pass the cards through 602A to calculate ... Details of this step are given in Section 3." Section 3 is two pages in length and has one page of instructions followed by a figure showing the wiring diagram for the control panel. I have quite clear memories of having to wire 602A control panels.
The inversion of a matrix of order 25 using this equipment took 75 hours.
Orange Coast College, Costa Mesa, California has some information, including a picture of a control panel for finding square roots, on the 602.