does multiplying a trigonometric functions in 2×2 matrix by a 2×1 matrix output a 1×2 matrix or a 2×1 matrix?

Question

the figure comes from this post

if I consider this vector

$$\left( \begin{array}{c} a_{1} \\ a_{2} \\ \end{array} \right)$$

as a 2 by 1 matrix

is the result on the right of the equation a 2 by 1 matrix which looks like

$$\left( \begin{array}{c} a_{2} \\ -a_{1} \\ \end{array} \right)$$

or a 1 by 2 matrix which looks like

$$\left( \begin{array}{c} a_{2}& -a_{1} \end{array} \right)$$

Answer 1

Multiplying a $\color{red}2\times \color{blue}2$ matrix ($\color{red}2$ rows and $\color{blue}2$ columns) by a $\color{blue}2\times1$ matrix ($\color{blue}2$ rows and $1$ column) makes sense because the blue $\color{blue}2$'s are equal, and the result is a $\color{red}2\times 1$ matrix ($\color{red}2$ rows and $1$ column).

The mnemonic that I learned back in the day for working out the dimensions of matrices in a product looks like this: $$ "(\color{red}2\times \color{blue}2)\cdot (\color{blue}2\times1) = \color{red}2\times 1" $$ where the inner $\color{blue}2$'s are equal (otherwise the product doesn't make sense), and cancel out, while the outer $\color{red}2$ and $1$ stay where they are.

Answer 2

The result is $\left( \begin{array}{c} a_{2} \\ -a_{1} \\ \end{array} \right).$