Linear algebra inverse of a matrix

In summary, the given matrix is its own inverse and can be verified by multiplying it by itself to get the identity matrix.
  • #1
Mdhiggenz
327
1

Homework Statement



The matrix given is
row 1: [0,1]
row 2: [1,0]

The matrix above if you switch row 1 with row 2 is just the identity matrix. So wouldn't that matrix already be the inverse of the identity matrix?



Homework Equations





The Attempt at a Solution

 
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  • #2
Mdhiggenz said:

Homework Statement



The matrix given is
row 1: [0,1]
row 2: [1,0]

The matrix above if you switch row 1 with row 2 is just the identity matrix. So wouldn't that matrix already be the inverse of the identity matrix?



Homework Equations





The Attempt at a Solution


What does switching rows have to do with being the inverse of anything? The inverse of the identity matrix is the identity matrix not your given matrix. Are you trying to find the inverse of the given matrix?
 
  • #3
Yes, I'm trying to find the inverse of the given matrix, however I ended up getting the same matrix I originally started with.
 
  • #4
Mdhiggenz said:
Yes, I'm trying to find the inverse of the given matrix, however I ended up getting the same matrix I originally started with.
Which says that your original matrix is its own inverse. You can easily verify that this is the case.
 
  • #5
Thanks for the replies guys, would this be a valid verification?

14dh8qc.jpg
 
  • #6
That's your original derivation and it's fine. But if M is your original matrix, by checking it mark44 just meant to test whether M*M=identity. It's easier to check whether an inverse is correct than to rederive it.
 
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  • #7
Mdhiggenz said:
Thanks for the replies guys, would this be a valid verification?

14dh8qc.jpg

That's an awful lot of work for that matrix. To find the inverse, just swap the two rows in your augmented matrix.

$$ \begin{bmatrix} 0 & 1 & | & 1 & 0 \\ 1 & 0 & | & 0 & 1\end{bmatrix} $$
$$ \equiv \begin{bmatrix} 1 & 0 & | & 0 & 1 \\ 0 & 1 & | & 1 & 0\end{bmatrix} $$
 
  • #8
That's what I did originally, but I wasn't sure if I could use that to verify.
 
  • #9
Finding the inverse and verifying that one matrix is the inverse of another are two different things. To find the inverse, use an augmented matrix and do row operations. To verify that one matrix is the inverse of another, just multiply the two matrices - if you get the identity matrix, that is verification that the two matrices you multiplied are inverses.

It's usually a lot less work to verify that two matrices are inverses than to find the inverse of a matrix.
 

FAQ: Linear algebra inverse of a matrix

1. What is the inverse of a matrix?

The inverse of a matrix is a matrix that when multiplied with the original matrix, gives the identity matrix as the result. In other words, it is a matrix that "undoes" the original matrix.

2. How do you find the inverse of a matrix?

The inverse of a matrix can be found by using various methods such as Gaussian elimination, LU decomposition, or using the adjugate matrix method. These methods involve performing a series of mathematical operations on the original matrix to obtain its inverse.

3. When does a matrix have an inverse?

A matrix has an inverse if and only if its determinant is non-zero. If the determinant is zero, the matrix is said to be singular and does not have an inverse.

4. Why is finding the inverse of a matrix useful?

Finding the inverse of a matrix is useful in many applications such as solving systems of equations, calculating the inverse of transformations, and solving optimization problems. It also allows for the simplification of mathematical expressions involving matrices.

5. Can all matrices be inverted?

No, not all matrices can be inverted. As mentioned earlier, a matrix must have a non-zero determinant in order to have an inverse. Matrices that have a determinant of zero are singular and do not have an inverse.

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