Composition: the matrix nearest the vector acts first

If $S$ applies first then $T$, the composite on column vectors is $TS$, so $(TS)\mathbf{x} = T(S\mathbf{x})$ . This is the standard graphics bug source: write the first transformation closest to the vector.

$(AB)\mathbf{x}$ means apply $B$, then $A$. Matrix multiplication is associative because composition of functions is associative. Parentheses move freely without changing the answer.

If $A$ is $m\times n$ and $B$ is $n\times p$, then $AB$ is $m\times p$: inner dimensions match at $n$. If rotation $R$ applies first and shear $S$ second, the matrix acting on $\mathbf{x}$ on the left is $SR\mathbf{x}$.

Function notation $f(g(x))$ matches matrix notation $FG\mathbf{x}$ when $F$ represents $f$ and $G$ represents $g$ consistently with column vectors .

Check your understanding. The tasks below rest on these ideas: Correct: $(AB)\mathbf{x}=A(B\mathbf{x})$, so $B$ acts on $\mathbf{x}$ first and $A$ second. Not quite: reading left-to-right reverses the order, the maps are sequential not simultaneous, and the determinant plays no role in ordering. Correct: matrices represent maps, and $(f\circ g)\circ h = f\circ(g\circ h)$ for functions. Not quite: transposes do not generally commute, the determinant is multiplicative not linear, and not every matrix is invertible. Correct: the inner dimensions $n$ match and contract, leaving the outer dimensions $m$ and $p$. Not quite: the other shapes mismatch or transpose the surviving dimensions. Correct: the first map sits nearest $\mathbf{x}$, so $R$ is on the right and $S$ on the left: $SR\mathbf{x}$. Not quite: $RS$ reverses the order, composition is multiplication not addition, and no inverse appears.