Exercise 3.4.6

Question

Let the reduced row echelon form of $A$ be 

$$\begin{pmatrix} 
1 & -3 & 0 & 4 & 0 & 5\
0 & 0 & 1 & 3 & 0 & 2 \
0 & 0 & 0 & 0 & 1 & -1 \
0 & 0 & 0 & 0 & 0 & 0
\end{pmatrix}$$

Determine $A$ if the first, third, and sixth colum of $A$ are 
$$\begin{pmatrix} 
1 \
-2 \
-1 \
3 
\end{pmatrix}
\begin{pmatrix} 
-1 \
1 \
2 \
-4 
\end{pmatrix}
\begin{pmatrix} 
3 \
-9 \
2 \
5 
\end{pmatrix}.$$

Jephian C.-H. Lin · Accepted Answer

Let $R$ be the matrix in reduced echelon form. We know that there is an invertible matrix $C$ such that $CA=R$. But now we cannot determine what $C$ is by the given conditions. However we know that the second column of $R$ is $-3$ times the first column of $R$. This means 
$$0=R\begin{pmatrix}3\1\0\0\0\end{pmatrix}=CA\begin{pmatrix}3\1\0\0\0\end{pmatrix}.$$
Since $C$ is invertible, we know that 
$$A\begin{pmatrix}3\1\0\0\0\0\end{pmatrix}=0.$$
And this means the second column of $A$ is also $-3$ times the first column of $A$. And so the second column of $A$ is $(-3,6,3,-9)$. Similarly we have that 
$$A\begin{pmatrix}-4\0\-3\1\0\0\end{pmatrix}=0=A\begin{pmatrix}-5\-2\0\0\1\1\end{pmatrix}$$
and get the answer that matrix $A$ is 
$$\begin{pmatrix}1&-3&-1&1&0&3\-2&6&1&-5&1&-9\-1&3&2&2&-3&2\3&-9&-4&0&2&5\end{pmatrix}.$$

Exercise 3.4.6

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Exercise 3.4.6

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