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Let $\mathbf{A}$ be a matrix, and let $\mathbf{x}$ and $\mathbf{y}$ be linearly independent vectors such that \[\mathbf{A} \mathbf{x} = \mat

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Let A be a matrix, and let x and y be linearly independent vectors 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 the ordered pair (a,b).

Mar 1, 2019

#1
+23071
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Let $$\mathbf{A}$$ be a matrix, and let $$\mathbf{x}$$ and $$\mathbf{y}$$ be linearly independent vectors such that
$$\mathbf{A} \mathbf{x} = \mathbf{y}, \mathbf{A} \mathbf{y} = \mathbf{x} + 2\mathbf{y}$$
Then we have that
$$\mathbf{A}^{-1} \mathbf{x} = a \mathbf{x} + b\mathbf{y}$$
or some scalars a and b.

Find the ordered pair (a,b).

$$\begin{array}{|rcll|} \hline Ax &=& y \quad | \quad \cdot A^{-1} \\ A^{-1}Ax &=& A^{-1}y \quad | \quad A^{-1}A = I \\ x &=& A^{-1}y \\ \mathbf{A^{-1}y} &\mathbf{=}&\mathbf{x} \\ \hline \end{array}$$

$$\begin{array}{|rcll|} \hline Ay &=& x + 2y \quad | \quad \cdot A^{-1} \\ A^{-1}Ay &=& A^{-1}x + 2A^{-1}y \quad | \quad A^{-1}A = I \\ y &=& A^{-1}x + 2A^{-1}y \quad | \quad \mathbf{A^{-1}y=x} \\ y &=& A^{-1}x + 2x \\ \mathbf{A^{-1}x} &\mathbf{=}&\mathbf{-2x+y } \\ \hline \end{array}$$

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

Mar 1, 2019