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# How do I find this vector?

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Let $$\mathbf{A}$$ be a matrix such that $$\mathbf{A}^{-1} = \begin{pmatrix} 1 & 1 \\ 2 & x \end{pmatrix}.$$

for some value of x . What is the vector that $$\mathbf{A}$$ maps to $$\begin{pmatrix} 1 \\ 0 \end{pmatrix}?$$

Apr 15, 2020

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Let $$A$$ be a matrix such that $$\mathbf{A}^{-1} = \begin{pmatrix} 1 & 1 \\ 2 & x \end{pmatrix}$$
for some value of $$x$$ .
What is the vector that  maps to $$\begin{pmatrix} 1 \\ 0 \end{pmatrix}$$?

I assume:

$$\begin{array}{|rcll|} \hline Av &=& \dbinom10 \quad & | \quad \times A^{-1}\\ A^{-1}Av &=& A^{-1}\dbinom10 \quad & | \quad A^{-1}A = I \\ Iv &=& A^{-1}\dbinom10 \quad & | \quad Iv = v \\ v &=& A^{-1}\dbinom10 \\\\ v &=& \begin{pmatrix} 1 & 1 \\ 2 & x \end{pmatrix}\begin{pmatrix} 1 \\ 0 \end{pmatrix} \\\\ v &=& \begin{pmatrix} 1*1+1*0 \\ 2*1+x*0 \end{pmatrix} \\\\ \mathbf{v} &=& \mathbf{ \dbinom12 } \\ \hline \end{array}$$

Apr 16, 2020