Backward for matmul

The problem, from first principles

Backward for multiplication handled scalars, where "sensitivity to a depends on b" is a single number times a single number. linear, this curriculum's very first question, is built on matrix multiplication instead, input @ weight.T, and every gradient computed in linear_regression and classification's training loops secretly depended on getting THIS backward rule right, even though those tracks derived it by hand for their specific case rather than deriving the fully general matmul backward rule this question builds directly.

The genuinely new difficulty here, beyond Backward for multiplication's scalar case: the SHAPES have to work out. grad_output has the shape of the OUTPUT, C, but the gradients you need to return must match the shapes of the INPUTS, A and B, which are generally different shapes entirely. Getting the transpose placement right is what makes those shapes line up correctly.

From theory to code

Theory derives dL/dA and dL/dB for C = A @ B by generalizing the scalar chain rule to matrices, landing on a formula built entirely from matrix multiplications and transposes, the exact operations 03-matrix-multiplication and Transpose, and its role in reshaping without copying data (both Math & Statistics) already cover individually.

Implement matmul_backward(grad_output, a, b) against that reasoning. The signature and docstring are already in the editor.

Constraints

• a is (m, k), b is (k, n), grad_output is (m, n) (matching C's shape).
• grad_a must come out shaped (m, k) (matching a), grad_b shaped (k, n) (matching b).
• Use only @ and .T, no explicit element-by-element loop.

Hints

Open one at a time. Each gives away a little more than the last.