LU Decomposition is a way to break a matrix down (factor) into the product of two matrices: one lower triangular and the other upper triangular. This LU product can be used to solve systems in the form 
. Of course, the first question that should come to mind is why do we care? We have a perfectly acceptable method (Gauss-Jordan) to solve systems. Why do we need an extra method that entails matrix multiplication and computing inverses in addition to row reductions? It turns out that if we have many 
vectors that we need to solve the system for, then LU decomposition beats Gauss-Jordan in computational efficiency. We will take the following matrix as an example:
To begin with, if we say 
how do we find the 
and 
matrices? First we row reduce to find an upper triangular matrix:
To find 
we can say 
. So we need to find 
by row reducing 
to 
:
Next we apply the definition above to find 
Now before moving on to solve a system, let's check our factorization: 
So, doing the multiplication shows that this is the correct 
factorization. Now let's try to solve the following system:
We can do this by saying 
. We will say that 
so that 
. Next we will solve for 
.
To solve for
we will use the inverse method so that 
. Again, to find 
we will row reduce:
Now we will solve 
with 
where we luckily already have 
from previous operations:
And this is the final solution of the 
system. Though this took the longest of any method, it is never practically implemented by hand. As mentioned before, its strengths lie in computing 
for the same 
over many 
vectors.
. Of course, the first question that should come to mind is why do we care? We have a perfectly acceptable method (Gauss-Jordan) to solve systems. Why do we need an extra method that entails matrix multiplication and computing inverses in addition to row reductions? It turns out that if we have many vectors that we need to solve the system for, then LU decomposition beats Gauss-Jordan in computational efficiency. We will take the following matrix as an example: how do we find the and matrices? First we row reduce to find an upper triangular matrix: we can say . So we need to find by row reducing to : factorization. Now let's try to solve the following system:. We will say that so that . Next we will solve for .To solve for
we will use the inverse method so that . Again, to find we will row reduce: with where we luckily already have from previous operations: system. Though this took the longest of any method, it is never practically implemented by hand. As mentioned before, its strengths lie in computing for the same over many vectors.