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# Let be a matrix, and let and be vectors such that neither is a scalar multiple of the other and such that Then we have that for some scalars

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Let $$\mathbf{A}$$ be a matrix, and let $$\mathbf{x}$$ and $$\mathbf{y}$$ be vectors such that neither is a scalar multiple of the other and such that $$\mathbf{A} \mathbf{x} = \mathbf{y}, \mathbf{A} \mathbf{y} = \mathbf{x} + 2\mathbf{y}.$$Then we have that $$\mathbf{A}^{-1} \mathbf{x} = a \mathbf{x} + b\mathbf{y}$$for some scalars a and b. Find a and b.

Any help would be appreciated, thank you very much!

Mar 18, 2020

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Let $$A$$ be a matrix, and let $$x$$ and $$y$$ be vectors such that neither is a scalar multiple of the other and such that
$$\mathbf{A} \mathbf{x} = \mathbf{y},\ \mathbf{A} \mathbf{y} = \mathbf{x} + 2\mathbf{y}$$.
Then we have that $$\mathbf{A}^{-1} \mathbf{x} = a \mathbf{x} + b\mathbf{y}$$ for some scalars $$a$$ and $$b$$.
Find $$a$$ and $$b$$.

$$\begin{array}{|rcll|} \hline \mathbf{A} \mathbf{x} &=& \mathbf{y} \quad &| \quad \times \mathbf{A}^{-1} \\ \mathbf{A}^{-1}\mathbf{A}\mathbf{x} &=& \mathbf{A}^{-1} \mathbf{y} \quad &| \quad \mathbf{A}^{-1}\mathbf{A} = \text{Identity matrix} \\ \mathbf{x} &=& \mathbf{A}^{-1}\mathbf{y} \\\\ \hline \mathbf{A} \mathbf{y} &=& \mathbf{x} + 2\mathbf{y} \quad &| \quad \times \mathbf{A}^{-1} \\ \mathbf{A}^{-1}\mathbf{A}\mathbf{y} &=& \mathbf{A}^{-1}\mathbf{x} + 2\mathbf{A}^{-1} \mathbf{y} \quad &| \quad \mathbf{A}^{-1}\mathbf{A} = \text{Identity matrix} \\ \mathbf{y} &=& \mathbf{A}^{-1}\mathbf{x} + 2\mathbf{A}^{-1} \mathbf{y} \quad &| \quad \mathbf{A}^{-1}\mathbf{y} = \mathbf{x} \\ \mathbf{y} &=& \mathbf{A}^{-1}\mathbf{x} + 2\mathbf{x} \\ \mathbf{A}^{-1}\mathbf{x} &=& -2\mathbf{x}+\mathbf{y} \\ \hline \end{array}$$

$$\mathbf{a=-2,\ b=1}$$

Mar 18, 2020
edited by heureka  Mar 18, 2020
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Thank you so much for showing your work!

rubikx2910  Mar 19, 2020