Finding the inverse of a 3x3 matrix

[nacho]

Hi, I had a question regarding the thread title.
I was introduced to this concept recently, and i was initially taught that this was done through elementary row operations. Which i found annoying..
Then Istumbled across another method, as can be seen here:
http://www.wikihow.com/Inverse-a-3X3-Matrix
I was just wondering whether this only worked for a matrix with a determinant of 1, or whether it could be used for any matrix (with the exception of det=0)

I find elementary row operations tedious and prone to error :|

[brightsky]

This is actually the more complicated way. The easier way (though a lot less methodical) is known as cofactor expansion.

[/0]

The method in the link uses Cramer's rule to find the inverse of a matrix.

The 'row operation' way of doing it is to have an augmented matrix

Then row reduce the augmented matrix until you get , then

Both methods work on any matrix with non-zero determinant.

[kamil9876]

The best and most general way (works for all n by n matrices) is a method based on elementary matrices: http://tutorial.math.lamar.edu/Classes/LinAlg/FindingInverseMatrices.aspx

The method tells you whether it is invertible or not and and if so gives the inverse.

[TrueTears]

Yeah there are many many ways, elementary row operations, cramer's rule, co factor expansions etc etc, you could also find the determinant of the matrix combinatorially haha

[nacho]

This is actually the more complicated way. The easier way (though a lot less methodical) is known as cofactor expansion.

Oh, I actually found this way quite a lot easier than row operations, i haven't heard of cofactor expansion before (don't even know what a cofactor is ) but it's mentioned in the method shown in the link right?
I'm having a link at Kamil's way.

[TrueTears]

Cofactor expansion is a recursive technique, it works nice for 3x3 but when you go to 4x4, 5x5 it becomes almost computationally impossible. You would then go about finding the inverse through where Adj(A) is the adjoint of the matrix A.

[brightsky]

This is actually the more complicated way. The easier way (though a lot less methodical) is known as cofactor expansion.

Oh, I actually found this way quite a lot easier than row operations, i haven't heard of cofactor expansion before (don't even know what a cofactor is ) but it's mentioned in the method shown in the link right?
I'm having a link at Kamil's way.

Actually, my bad. Cofactor expansion is for finding the determinant, and so would probably make life easier when using Cramer's rule. If I'm not mistaken, kamil's link is just the "row operation" way?

The row operation way is defs the easier, less long-winded method.

[nacho]

This is actually the more complicated way. The easier way (though a lot less methodical) is known as cofactor expansion.

Oh, I actually found this way quite a lot easier than row operations, i haven't heard of cofactor expansion before (don't even know what a cofactor is ) but it's mentioned in the method shown in the link right?
I'm having a link at Kamil's way.

Actually, my bad. Cofactor expansion is for finding the determinant, and so would probably make life easier when using Cramer's rule. If I'm not mistaken, kamil's link is just the "row operation" way?

The row operation way is defs the easier, less long-winded method.

Each to their own i guess, i find row-operation more difficult and slower, I just needed the thumbs up on whether i could use Cramer's rule for all scenarios. (Except obviously for a zero determinant).
Just a little confused here: http://mathforum.org/library/drmath/view/55480.html
It says: "If the determinant is zero, you know your matrix is singular (it has
no inverse). If the determinant is one (unitary matrix), the inverse
of your matrix will be what is called the "adjoint of A" (denoted by
adj(A)). Note that the adjoint of an NxN matrix is also an N x N
matrix."

I haven't gone over this at school , so I don't really know what the adjoint of A (any matrix) is, but i know it is found by switching the rows so they become the collumns (hopefully you can understand what i mean) as seen in the link i provided. However, in the quote from the dr. math website, what does it mean 'if the determinant is one (unitary matrix), the inverse of your matrix will be what is called the "adjoing of A" (denotes by adj(A))'
So does this mean that for a matrix with a determinant of 1, you can easily calculate the inverse, by just finding the adjoint of that matrix? Or am i mistaken.

So cofactor expansion is used to find the determinant, which is the method where you select a row or column in the 3x3 matrix, and then using that row you find the determinants of the minor matrix and then do those calculations..

This is used in both Cramer's rule and row operations to find the determinant, correct?

[TrueTears]

Row operations is the easiest I reckon, again Cramer's rule for 3x3 is not bad, but for 4x4 or 5x5 or n x n matrices in general, you will need to compute n+1 determinants.

[nacho]

Row operations is the easiest I reckon, again Cramer's rule for 3x3 is not bad, but for 4x4 or 5x5 or n x n matrices in general, you will need to compute n+1 determinants.

Ohwell, who needs to manually solve a 4x4 or 5x5 matrix, that's insane! I'm only required to master the 3x3 matrix .

[TrueTears]

haha you'll see that after learning some linear algebra there are very slick ways of finding inverses, eg, you could reduce your matrix into row echelon form (which is upper triangular), then find the determinant by finding the product of the diagonal. Or you could use a combination of row reductions and make a row or column have as many 0's as possible then cofactor expand it with ease.

[dcc]

My personal favourite (for invertible matrices without 0 as an eigenvalue) is to use the Cayley-Hamilton theorem and factorize.  Though this only works sometimes, you still need to calculate eigen-things (on the other hand, it gives you a 'clean' expression for the inverse).

[Mao]

Gaussian elimination (row operations) works most of the times, but it tend to bring on numerical errors for larger systems. Iterative methods are the way to go to solving larger systems.

[kamil9876]

for invertible matrices without 0 as an eigenvalue

Does any invertible matrix have 0 as an eigenvalue?