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Do you know how to use row reduction?
Write up a matrix with your entries, draw a line between your entries and the entries of an identity matrix.
Use row reduction on the entire system until the left-hand side becomes an identity matrix - the right-hand side will be your inverse.
The inverse can also be found by finding the adjoint of your matrix and multiplying it by the inverse of the determinant of the original matrix.
Do you know how to do any of this?
Could you show me?
I wish I could - I have to leave. If you don't want to use the adjoint method (i.e. just use the first, which shouldn't be that bad seeing as your matrix is small), go to this website http://www.khanacademy.org/ and look up "inverse matrix" - there are three parts. He will show you (they're just YouTube clips) the method I was talking about before. If you still need help in several hours, I'll see what I can do. Good luck - it's just a procedure.
I just punched out an answer in the last couple of minutes - it shouldn't take you that long. Just be sure of your algebra skills and don't get flustered.
I'll post it later if you need it (there are exponentials everywhere and you can't write up matrices properly on this thing).
PS - always check that what you have *is* the inverse by multiplying it with your original matrix - you should end up with the identity matrix.
Hey, js14, I don't know how you went with your question, but I had a look during the day. I did it using the adjoint way. I can't write out the solution on this site, but if you want it, let me know; I'll scan and attach it here.
sure I would appreciate that.
I don't know how useful it will be since it doesn't explain what's going on. What I did was just find the general solution for the inverse 2 x 2 square matrix (so you'll see a matrix with a b c d at the top. I then just took the 'minors', then the cofactors, then the cofactor matrix and from that, the adjoint matrix. Multiplying the adjoint with the determinant of the original matrix gives you the inverse. I then just plugged in a = e^2, b=-e^x, c=e^(2x) and d=e^(3x). Enjoy! If there are any problems, let me know.