Matrix multiplication

A 2 × 2 matrix is a linear map of the plane, and its columns are where it sends $\hat\imath = (1, 0)$ and $\hat\jmath = (0, 1)$. The product $AB$ is the matrix of doing two maps in a row, first $B$ and then $A$: $(AB)x = A(Bx)$ for every vector $x$. The matrix next to $x$ acts first, so a product is read from right to left.

In the order $AB$, the left picture shows $B$ on the standard grid: $\hat\imath$ and $\hat\jmath$ as faint dashed arrows, and their images $b_1$ and $b_2$, the columns of $B$, in blue. The right picture shows $A$: the grid it makes of the standard grid, its columns $a_1$ and $a_2$ in ink, and the columns of the product, $Ab_1$ and $Ab_2$, in blue. The flag shows the same two steps on a shape. Each picture has its own scale, because a product can reach far beyond its factors.

Where does a column of $AB$ come from? $B$ sends $\hat\imath$ to $b_1 = (b_{11}, b_{21})$, which is $b_{11}$ steps of $\hat\imath$ and $b_{21}$ steps of $\hat\jmath$. $A$ is linear, so it sends that point to the same steps of its own columns: $b_{11}$ of $a_1$ followed by $b_{21}$ of $a_2$. It only reads the two numbers of $b_1$, not how $b_1$ was made. On the right the selected product column is drawn as this recipe, two thin legs along $a_1$ and $a_2$ that end at the blue arrow. Drag $b_1$ on the left and the legs stretch and shrink with its numbers.

The rule taught as row times column is this recipe written in coordinates. The x-coordinate of $Ab_j$ adds up the horizontal parts of its two legs, $a_{11}b_{1j} + a_{12}b_{2j}$: row 1 of $A$ against column $j$ of $B$. The y-coordinate comes from row 2 in the same way. Row × column draws these parts along the axes, each labelled with its product.

Order matters because the second map works on what the first one produced. Turn by a quarter and then shear: $\hat\imath$ goes to $(0, 1)$ and then to $(1, 1)$. Shear first and then turn: $\hat\imath$ stays at $(1, 0)$ and then goes to $(0, 1)$. Once the other order has been composed too, its result appears on the right as dashed blue arrows, until A or B changes. Some pairs do commute. Two turns of the plane about the origin add their angles in either order.

Area factors multiply. $B$ turns the unit square into a parallelogram of area $|\det B|$, and $A$ then multiplies every area by $|\det A|$. With signs, $\det(AB) = \det A\cdot\det B$, and because numbers commute, $\det(AB) = \det(BA)$ even when $AB \ne BA$. In the default, $\det A = 2$ and $\det B = 3$, so both products have determinant 6.