Post by: nacho on November 22, 2010, 10:21:19 pm
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 :|
Post by: brightsky on November 22, 2010, 10:30:02 pm
Post by: /0 on November 22, 2010, 10:36:52 pm
The 'row operation' way of doing it is to have an augmented matrix
Then row reduce the augmented matrix until you get
Both methods work on any matrix with non-zero determinant.
Post by: kamil9876 on November 22, 2010, 10:43:57 pm
The method tells you whether it is invertible or not and and if so gives the inverse.
Post by: TrueTears on November 22, 2010, 10:51:42 pm
Post by: nacho on November 22, 2010, 10:55:08 pm
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.
Post by: TrueTears on November 22, 2010, 10:57:28 pm
Post by: brightsky on November 22, 2010, 10:58:14 pm
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.
Post by: nacho on November 22, 2010, 11:12:59 pm
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).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.
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?
Post by: TrueTears on November 22, 2010, 11:19:05 pm
Post by: nacho on November 22, 2010, 11:21:01 pm
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 :).
Post by: TrueTears on November 22, 2010, 11:23:51 pm
Post by: dcc on November 24, 2010, 04:36:28 pm
Post by: Mao on November 24, 2010, 04:56:26 pm
Post by: kamil9876 on November 24, 2010, 07:47:25 pm
Quote
for invertible matrices without 0 as an eigenvalue
Does any invertible matrix have 0 as an eigenvalue? ;)
Post by: /0 on November 24, 2010, 09:03:48 pm
I just discovered this recently lol
Post by: dcc on November 24, 2010, 09:06:47 pm
Quotefor invertible matrices without 0 as an eigenvalue
Does any invertible matrix have 0 as an eigenvalue? ;)
I'm a distinguished member of the accidental tautology club :D (perhaps they should call it the idiot club)
Post by: kamil9876 on November 24, 2010, 09:07:21 pm
2 0
0 2
has determinant 4=2*2, because the eigenvalue 2 appears "twice".
Post by: kamil9876 on November 24, 2010, 09:08:08 pm
Quotefor invertible matrices without 0 as an eigenvalue
Does any invertible matrix have 0 as an eigenvalue? ;)
I'm a distinguished member of the accidental tautology club :D
Nice, I already knew that you either were or weren't.
Post by: dcc on November 24, 2010, 09:12:20 pm
Post by: Tan on November 28, 2010, 10:34:03 pm
Post by: TrueTears on November 29, 2010, 01:50:17 am
Wow I didn't know finding the inverse of a 3x3 matrix was even possible. My own methods teacher said it can't be done lol.lol he didn't want to scare you guys
Post by: pi on November 29, 2010, 01:46:16 pm
Wow I didn't know finding the inverse of a 3x3 matrix was even possible. My own methods teacher said it can't be done lol.lol he didn't want to scare you guys
too late ;)