Linear Algebra: Inverse of Elementary Matrix

[ephemeral1]

Homework Statement

What is the inverse of this elementary matrix? 2x2 matrix
1 0
3 -1


E=

Homework Equations

None

The Attempt at a Solution

This is what I think:
1 0
-3 -1

Is that right? Or do you leave the 3 positive? If so, please explain. Thank you.

 

[LCKurtz]

Homework Statement

What is the inverse of this elementary matrix? 2x2 matrix
1 0
3 -1


E=

Homework Equations

None

The Attempt at a Solution

This is what I think:
1 0
-3 -1

Is that right? Or do you leave the 3 positive? If so, please explain. Thank you.

Do you get the identity matrix when you multiply your inverse by the original?

 

[ephemeral1]

No, I did not. But when I multiply
1 0
3 -1 by the original, I get the identity matrix. I don't understand why the 3 is positive. Don't we have to change the sign on additional constant? I thought the 3 is the additional constant? Or is it because the 1 on the second row is negative, we don't have to change the sign on the -3 when doing the inverse? Please explain. Thank you.

 

[LCKurtz]

What method did you use to calculate the inverse? That should answer your question.

 

[rsa58]

you are correct you must change the sign of 3, however the determinant is equal to -1. remember you have to divide by the determinant when using cramer's rule.

 

[rsa58]

another way to calculate the inverse is to write the matrix equation AX=I and solve for the unknowns in X.