Considering the following two matrices I am trying to find for what values of k are the two matrices row equivalent. The hint I received is "if we row reduce B to make a zero row, we can more easily see the value of k".
$$
A =
\begin{pmatrix}
1 & 0 & 1 \\
0 & 1 & 2 \\
0 & 0 & 0 \\
\end{pmatrix} \hspace{1.0cm}
B =
\begin{pmatrix}
1 & 0 & 1 \\
0 & 2 & 4 \\
2 & k & 6 \\
\end{pmatrix}
$$
For matrix B I started by dividing row 2 elements by 2 and row 3 elements by 3 to get the following:
$$
B =
\begin{pmatrix}
1 & 0 & 1 \\
0 & 1 & 2 \\
1 & k & 3 \\
\end{pmatrix}
$$
Then, I made row 3 equal to subtracting row 1 from row 3 to produce the following:
$$
B =
\begin{pmatrix}
1 & 0 & 1 \\
0 & 1 & 2 \\
0 & k-1 & 0 \\
\end{pmatrix}
$$
And so B is inconsistent for k = 1 if the resulting is not equal to 0 but consistent if it is. For example, 0 + 0 + 0 = 0 is fine but 0 + 0 + 0 = 7 is not.
How do I go about finding the values for k that make the two matrices row equivalent? Thank you.
Best Answer
When you divided $R_3$ by $2$ you didn't divide $k$ so you reduction is wrong.
Reducing $B$ you obtain : $$ \begin{pmatrix} 1 & 0 & 1 \\ 0 & 2 & 4 \\ 2 & k & 6 \\ \end{pmatrix} \to \begin{pmatrix} 1 & 0 & 1 \\ 0 & 1 & 2 \\ 0 & k-2 & 0\\ \end{pmatrix}$$
So $ A \sim {B}$ for $k=2$. No other value is possible because when $k \neq 2$ the third row isn't reducible to $(0,0,0)$ using any elementary operation.