Is there a certain strategy or way to solve these because they take a really long time. Im sure this stratgey would also work for inverse matricies.

row reduction, when the left columns become the Identity matrix the system is solved.

there's really no other way to do it, in practice you always have a calculator or computer do it for you.

yea i know that, but they take a while, and for my test on monday my teacher is going to make us do it by "hand", so thats why i asked

ah, for that you just need to take it slow. and check each step. A single mistake can make it so that the entire thing is bunk.

Row-reduction is pretty much the only way of solving these.

I second that. Gaussian or Gaussian-Jordan Elimination is the only way.

To say Gauss-Jordan elimination is the only way is a bit strong, to say the least. I rarely use Guass-Jordan elimination, and I deal with matrices a lot. I frequently use various decomposition techniques: LU, Cholesky, QR, SVD, ...

 

Guass-Jordan elimination is the only way to do it by hand, perhaps.

Yes, I think that's probably the best thing to say. I don't think that people would generally ask for one to do LU decomposition on a matrix by hand in exam conditions

LU Decomposition is a mulitple of lower triangle and upper triangle matrices, so it does not produce the reduced matrix.

it is however a mehtod for solving systems of equations...

 

which is what the poster originally asked, personally I don't see how it really benfits the user as you can solve an upper triangular matrix U using backsubstitution.

 

maye I' missing part of its ability.

Speaking computationally, straight LU decomposition usually doesn't save much at all. But, there are incomplete LU decomposition algorithms that are much faster and usually give results that can be tuned in accuracy (whether you want 0.1% error or 0.001% or something else, for example). These are used for a lot of computational simulations -- especially when the linear system is an approximation of a non-linear system. In that case, you do not need to solve the linear system to really high accuracy (like a complete LU or Gaussion Elimination would), because that approximated linear solution is going to be used as the latest guess in the iterative method needed to solve the nonlinear problem.