In my class we've been talking about elementary row operations of matrices, for example, to get its inverse. I remember the teacher saying, those operations don't work for columns; however, in my book, it says they do work. Can I use them on columns to get to the identity matrix for instance, so that I can, then, get its inverse.
$\begingroup$ $\endgroup$
Add a comment |
1 Answer
$\begingroup$ $\endgroup$
3 The Gauss-Jordan method is based on rows operations by
$$A^{-1}\cdot (A|I)=(I|A^{-1})$$
we can also operate by columns operation that is
$$\begin{pmatrix}A\\I\end{pmatrix}A^{-1}=\begin{pmatrix}I\\A^{-1}\end{pmatrix}$$
- $\begingroup$ I haven't seen both of those equations. Is the first one the same as $A.A^{-1}=I=A^{-1}.A$? If so, I understand that for that operation, the rows should be used and the columns can't? What is the second equation then? $\endgroup$Duarte Arribas– Duarte Arribas2019-10-24 22:22:04 +00:00Commented Oct 24, 2019 at 22:22
- $\begingroup$ @DuarteArribas Since rows operation are equivalent to left matrix multiplicationwe can summarize the method by $A^{-1}\cdot (A|I)=(I|A^{-1})$. We can of course operate by columns operations which correspond to roght matrix multiplication. For more details refer also to this OP. $\endgroup$user– user2019-10-24 22:28:00 +00:00Commented Oct 24, 2019 at 22:28
- $\begingroup$ @DuarteArribas To emphasize the point here, column operations work, too, but you have to augment the matrix by appending $I$ below it instead of to the right. $\endgroup$amd– amd2019-10-24 22:47:19 +00:00Commented Oct 24, 2019 at 22:47