Matrix Transpose 1x2

Question

Let's say I have a $1\times 2$ matrix $U$.

Working out $\text{transpose}(U) \times U$ and then $ U \times \text{transpose}(U). $

Why does one of these give a value while the other gives you a matrix.

Why does it matter where the transpose is?

Answer 1

Matrix multiplication is not commutative. In other words, you cannot switch around the order and get the same result.

For example, let's say you have $A$, which is a $3$ x $1$ matrix and $B$, which is a $2$ x $3$ matrix.

The product $AB$ is undefined, since the number of columns in $A$ must necessarily match up with the number of rows in $B$ (or else you couldn't compute the dot product between a row in $A$ and a column in $B$).

However, the product $BA$ is defined. It's a $2$ x $1$ matrix.

So this shows that it's simply not commutative, that the order in which you multiply matters. If thinking about this in a linear transformation sense, if you rotate and then shear, for example, this may not be the same as shearing and then rotating.