## Find a Matrix that Maps Given Vectors to Given Vectors

## Problem 44

Suppose that a real matrix $A$ maps each of the following vectors

\[\mathbf{x}_1=\begin{bmatrix}

1 \\

1 \\

1

\end{bmatrix}, \mathbf{x}_2=\begin{bmatrix}

0 \\

1 \\

1

\end{bmatrix}, \mathbf{x}_3=\begin{bmatrix}

0 \\

0 \\

1

\end{bmatrix} \]
into the vectors

\[\mathbf{y}_1=\begin{bmatrix}

1 \\

2 \\

0

\end{bmatrix}, \mathbf{y}_2=\begin{bmatrix}

-1 \\

0 \\

3

\end{bmatrix}, \mathbf{y}_3=\begin{bmatrix}

3 \\

1 \\

1

\end{bmatrix},\]
respectively.

That is, $A\mathbf{x}_i=\mathbf{y}_i$ for $i=1,2,3$.

Find the matrix $A$.

(*Kyoto University Exam*)

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