Ranked solutions of the matric equation A 1 X 1 = A 2 X 2

Let G F ( p z ) denote the finite field of p z elements. Let A 1 be s × m of rank r 1 and A 2 be s × n of rank r 2 with elements from G F ( p z ). In this paper, formulas are given for finding the number of X 1 , X 2 over G F ( p z ) which satisfy the matric equation A 1 X 1 = A 2 X 2 , where X 1 is m × t of rank k 1 , and X 2 is n × t of rank k 2 . These results are then used to find the number of solutions X 1 , …, X n , Y 1 , …, Y m , m , n > 1, of the matric equation A 1 X 1 … X n = A 2 Y 1 … Y m .