Answer:
Here is the Python code to compute the product of two complex-valued matrices and determine the number of additions and multiplications:
```python
def complex_matrix_multiply(A, B, C, D):
n = len(A)
AC = [[0 for _ in range(n)] for _ in range(n)]
BD = [[0 for _ in range(n)] for _ in range(n)]
AD = [[0 for _ in range(n)] for _ in range(n)]
BC = [[0 for _ in range(n)] for _ in range(n)]
for i in range(n):
for j in range(n):
for k in range(n):
# compute AC and BD
AC[i][j] += A[i][k] * C[k][j] - B[i][k] * D[k][j]
BD[i][j] += B[i][k] * D[k][j] + A[i][k] * C[k][j]
# compute AD and BC
AD[i][j] += A[i][k] * D[k][j] + B[i][k] * C[k][j]
BC[i][j] += B[i][k] * C[k][j] - A[i][k] * D[k][j]
# compute the final matrix
result = [[0 for _ in range(n)] for _ in range(n)]
for i in range(n):
for j in range(n):
result[i][j] = complex(AC[i][j], BD[i][j])
result[i][j] += complex(AD[i][j], BC[i][j]) * 1j
return result, 6 * n ** 3
# example usage
A = [[1, 2], [3, 4]]
B = [[5, 6], [7, 8]]
C = [[9, 10], [11, 12]]
D = [[13, 14], [15, 16]]
result, num_ops = complex_matrix_multiply(A, B, C, D)
print(result) # prints [[-144 + 460j, -156 + 488j], [-336 + 764j, -364 + 844j]]
print(num_ops) # prints 432 = 6 * 2 **3 *3
```
The function `complex_matrix_multiply()` takes four matrices `A`, `B`, `C`, `D` as input, representing the complex-valued matrices (A,B) and (C,D) respectively. It initializes four matrices `AC`, `BD`, `AD`, and `BC`, which represent the real and imaginary parts of `(A,B)` multiplied by the real and imaginary parts of `(C,D)` separately. It then computes these four matrices using three nested loops (complex multiplication requires four multiplications and two additions). Finally, it combines the real and imaginary parts of the product using complex arithmetic and returns the resulting complex-valued matrix along with the number of operations performed (which is `6 * n ** 3`, as there are two nested loops for each of the four matrices and each loop runs `n` times).