Question M40 in Section RREF

Student23

I am struggling to get the matrix B and the matrix C to row reduce to be identical. Could someone explain what row operations I need to do to get the matrices to be the same?

Matrix B:
$$
\left(\begin{matrix}1 & 3 & -2 & 2 \\ -1 & -2 & -1 & -1 \\ -1 & -5 & 8 & -3 \end{matrix}\right)
$$

Matrix C:
$$
\left(\begin{matrix}1 & 2 & 1 & 2 \\ 1 & 1 & 4 & 0 \\ -1 & -1 & -4 & 1 \end{matrix}\right)
$$

Thank you!

mark

Here's a quick idea, rather than a full solution. The reduced row echelon form of both matrices should be the same; let's call that matrix $R$. First, row reduce $B$ to $R$ via a sequence of operations, say
$$
E_1, E_2, \ldots, E_m.
$$
Then row reduce $C$ to $R$ via another sequence of operations, say
$$
F_1, F_2, \ldots, F_n.
$$
Finally, if you apply the sequence
$$
E_1, E_2, \ldots, E_m, F_n^{-1}, \ldots, F_2^{-1}, F_{1}^{-1}
$$
to $B$, you should get $C$.

---

Note that this basic idea is quite important. In fact, you'll need to use something similar in problem T11 on page 33 of our text. Here's another problem that's related.