While studying for the second exam in my ECE 874 course, I was reviewing the matrix calculus tables and I happened to notice several patterns in the equations.  From basic calculus, we know that there are simple shortcuts one may use to jump directly to an accurate answer, and I eventually found this to be no different.  First of all, I noticed that typically when the particular matrix targeted was not being transposed, the results were simply the transposed matrices on each side of that matrix.  Similarly, when that matrix targeted was being transposed, the results were simply all the original matrices (less the targeted matrix) switched.  Later trials on more complex examples were made quite simplified.  For example:

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