Two Layers, One Matrix

A network has two purely linear layers — no bias, no activation function. Layer $A$ runs first, then layer $B$ :

$A = \begin{bmatrix} 2 & 1 \\ 0 & 3 \end{bmatrix} \qquad B = \begin{bmatrix} 1 & -1 \\ 2 & 0 \end{bmatrix}$

So the network computes $x \mapsto B(Ax)$ .

Because nothing nonlinear sits between them, these two layers collapse into a single equivalent layer. Which single matrix $C$ satisfies $Cx = B(Ax)$ for every input $x$ ?

Two Layers, One Matrix

Hints

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Two Layers Collapse Into One