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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
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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).

 

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

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


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

 

laugh

 Mar 1, 2019

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