- #1

evinda

Gold Member

MHB

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The general solution of the system $Ax=\begin{bmatrix}

1\\

3

\end{bmatrix}$ is $x=\begin{bmatrix}

1\\

0

\end{bmatrix}+ \lambda \begin{bmatrix}

0\\

1

\end{bmatrix}$. I want to find the matrix $A$.

I have done the following so far:

$$x=\begin{bmatrix}

1\\

0

\end{bmatrix}+ \lambda \begin{bmatrix}

0\\

1

\end{bmatrix}=\begin{bmatrix}

1\\

\lambda

\end{bmatrix}.$$

Let $A=\begin{bmatrix}

a_{11} & a_{12}\\

a_{21} & a_{22}

\end{bmatrix}$.$$\begin{bmatrix}

a_{11} & a_{12}\\

a_{21} & a_{22}

\end{bmatrix} \begin{bmatrix}

1\\

\lambda

\end{bmatrix}=\begin{bmatrix}

1\\

3

\end{bmatrix} \Leftrightarrow \begin{bmatrix}

a_{11}+ \lambda a_{12}\\

a_{21}+\lambda a_{22}

\end{bmatrix}=\begin{bmatrix}

1\\

3

\end{bmatrix} \Leftrightarrow \begin{Bmatrix}

a_{11}+\lambda a_{12}=1 \\

a_{21}+\lambda a_{22}=3

\end{Bmatrix} \Leftrightarrow \begin{Bmatrix}

a_{11}=1-\lambda a_{12} \\

a_{21}=3-\lambda a_{22}

\end{Bmatrix}. $$So $A$ is the following matrix:

$$A=\begin{bmatrix}

1-\lambda a_{12} & a_{12}\\

3-\lambda a_{22} & a_{22}

\end{bmatrix}.$$

Is everything right? Can we get more information or is this sufficient? (Thinking)