Exercise 3: The A * B vs. A @ B NumPy Bug — Possible Solution ==================================================================== WHAT'S ACTUALLY HAPPENING ------------------------------ In NumPy, the * operator does NOT perform the row-by-column matrix multiplication defined in this chapter. It performs entry-wise (element-wise) multiplication instead - multiplying each entry of A by the matching entry in the same position in B, position by position. This is a completely different mathematical operation from real matrix multiplication, even though both A * B and the correct A @ B produce a matrix that looks superficially plausible. WHAT THE CODE SHOULD SAY INSTEAD ------------------------------ The teammate should use A @ B (or equivalently np.matmul(A, B)) to get genuine matrix multiplication - the row-by-column dot-product operation this chapter defines, where each output entry is the dot product of a row from A and a column from B. WHY THE BUG WASN'T CAUGHT BY AN ERROR OR CRASH ------------------------------ Both A and B are 3x3 matrices in this scenario, so entry-wise multiplication (A * B) and real matrix multiplication (A @ B) both produce a valid 3x3 result - same shape, same data type, no exception raised anywhere. NumPy has no way to know which operation the programmer actually intended; it silently computes whichever operator was written and returns a result. This is exactly why the output "looks wrong" only when compared against a hand calculation, rather than surfacing as a crash - the code runs successfully and produces real numbers, just not the mathematically correct ones for the intended operation. WHY THIS WORKS AS AN ANSWER ------------------------------ It identifies the specific operator confusion this chapter's own warning box names (* is entry-wise, @ is real matrix multiplication), gives the concrete fix, and explains the silent-failure mechanism directly in terms of both operators producing a same-shaped, type-valid result for square matrices of matching size - which is exactly why no exception was ever raised to reveal the mistake.