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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 :)

Apr 17, 2023

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

Apr 17, 2023
#2
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Do not post A O P S homework.  That is cheating.

Apr 17, 2023
#3
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Oh my goodness, I think I've figured it out.  Will it resemble A(x+y)?  Consequently, it would be x+3y.

Apr 19, 2023