Linear algebra inverse of a matrix

[Mdhiggenz]

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

 

[Dick]

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?

 

[Mdhiggenz]

Yes, I'm trying to find the inverse of the given matrix, however I ended up getting the same matrix I originally started with.

 

[Mark44]

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.

 

[Mdhiggenz]

Thanks for the replies guys, would this be a valid verification?

 

[Dick]

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.

 

[Mark44]

Thanks for the replies guys, would this be a valid verification?

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

$$ \begin{bmatrix} 0 & 1 & | & 1 & 0 \\ 1 & 0 & | & 0 & 1\end{bmatrix} $$
$$ \equiv \begin{bmatrix} 1 & 0 & | & 0 & 1 \\ 0 & 1 & | & 1 & 0\end{bmatrix} $$

 

[Mdhiggenz]

That's what I did originally, but I wasn't sure if I could use that to verify.

 

[Mark44]

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.