Let $\mathbf{A}$ and $\mathbf{B}$ be matrices, and let $\mathbf{x}$ and $\mathbf{y}$ be vectors such that neither is a scalar multiple of the other satisfying \[\mathbf{A} \mathbf{x} = \mathbf{y}, \mathbf{A} \mathbf{y} = \mathbf{x} + 2\mathbf{y}\]and \[\mathbf{B} \mathbf{x} = \mathbf{x} + \mathbf{y}, \mathbf{B} \mathbf{y} = 2\mathbf{y}.\]Then there exist scalars $a, b, c, d$ such that \begin{align*} (\mathbf{A}\mathbf{B})\mathbf{x} = a \mathbf{x} + b\mathbf{y},\\ (\mathbf{B}\mathbf{A})\mathbf{x} = c \mathbf{x} + d\mathbf{y}. \end{align*}Enter $a,b,c,d$ in that order.

So I'm confused because we don't know what x and y are exactly. For example, for finding (AB)x, we would do A(Bx). Bx=x+y and then Ax=y. So would it be y(x+y)? But then we are looking for scalars and there is no y^2 or vectors squared anywhere. Please help, any tips will be greatly appreciated :)

Saphia1123 Apr 17, 2023

#1**+1 **

OHHHHH I think I just figured it out. Would it be like A(x+y)? So then it would be x+3y?

Saphia1123 Apr 17, 2023