Can Column Operations be Used to Find the Inverse of a Matrix?

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In summary, the conversation discusses the use of row and column operations to find the inverse of a matrix. The participant suggests augmenting the matrix with the identity matrix and performing row operations, but is unsure if the same method can be applied using column operations. The expert clarifies that the process is essentially the same, but column operations use the columns of the matrix instead of the rows. The participant also asks about using both row and column operations simultaneously, to which the expert explains that it is possible algebraically by solving for the inverse.
  • #1
rock.freak667
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If I have a matrix

[tex]
A= \left(
\begin{array}{ccc}
1 & 2 & 3\\
0 & -1 & 4\\
1 & 1 & 6
\end{array}
\right)
[/tex]
and I need to find [itex]A^{-1}[/itex] I would just augment with the identity matrix and then do row operations. But if I want to use column operations instead does it work in the same manner? because I think if use the column operations, the matrix A would be reduced to RRE form but nothing will happen to the identity matrix.
(Not too sure if I was clear about my problem.)
 
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  • #2
what are column operations?
 
  • #3
You need to adjoin an identity matrix below the original, not to its right.
 
  • #4
rock.freak667 said:
I think if use the column operations, the matrix A would be reduced to RRE form
You mean reduced column echelon form.
 
  • #5
Hurkyl said:
You mean reduced column echelon form.
Yeah sorry about that.That is what I meant.
Hurkyl said:
You need to adjoin an identity matrix below the original, not to its right.

Below it? So there is no way to both row and column operations at the same time to find the inverse or whatever?Also how would I solve a system of equations using column operations only?
 
  • #6
wouldn't these column operations be the same thing as row reducing the transpose?
 
  • #7
ice109 said:
wouldn't these column operations be the same thing as row reducing the transpose?

It is the same thing basically except that column operations uses the columns of the matrix,
 
  • #8
rock.freak667 said:
Yeah sorry about that.That is what I meant.


Below it? So there is no way to both row and column operations at the same time to find the inverse or whatever?
Not with this particular bookkeeping method. Of course you can use both kinds of operations to find an inverse: algebraically, if R is the matrix denoting your row operations and C is the one denoting your column operations, then if you reduce your matrix to the identity, that says
RAC = I​
which you can easily solve for A.
 
  • #9
oh okay then thanks
 

Related to Can Column Operations be Used to Find the Inverse of a Matrix?

1. What are column operations in matrices?

Column operations are mathematical operations performed on the columns of a matrix. These include adding, subtracting, multiplying, and dividing columns, as well as swapping columns or scaling them by a constant.

2. What is the purpose of using column operations in matrices?

Column operations are used to manipulate a matrix in order to solve systems of equations, find inverses, and perform other operations on matrices. They can also help to simplify matrices and make them easier to work with.

3. How do you perform column operations on a matrix?

To perform column operations, you can use elementary row operations, which include multiplying a column by a constant, adding a multiple of one column to another, and swapping two columns. These operations can be done by hand or using a calculator or computer program.

4. Can you combine different types of column operations in a single matrix?

Yes, you can combine different types of column operations in a single matrix. However, the order in which you perform the operations can affect the final result, so it is important to follow the correct order of operations.

5. What are some real-world applications of column operations in matrices?

Column operations are used in various fields such as engineering, physics, economics, and computer science. They are particularly useful in solving systems of equations, analyzing data, and performing transformations in computer graphics.

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