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Homematrix inverse
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matrix inverse
1 Matrix Inverse
$AA^{{1}}=A^{{1}}A=I_{n}$ 
where $I_{n}$ is the $n\times n$ identity matrix.
It should be stressed that only square matrices have inverses proper– however, a matrix of any size may have “left” and “right” inverses (which will not be discussed here).
A precondition for the existence of the matrix inverse $A^{{1}}$ (i.e. the matrix is invertible) is that $\det A\neq 0$ (the determinant is nonzero), the reason for which we will see in a second.
The general form of the inverse of a matrix $A$ is
$A^{{1}}=\frac{1}{\det(A)}\operatorname{adj}(A)$ 
where $\operatorname{adj}(A)$ is the adjugate of $A$ (the matrix formed by the cofactors of $A$, i.e. with $\operatorname{adj}(A)_{{ij}}=C_{{ij}}(A)$).^{1}^{1}Some other sources call the adjugate the adjoint; however on PM the adjoint is reserved for the conjugate transpose. This can also be thought of as a generalization of the $2\times 2$ formula given in the next section. However, due to the inclusion of the determinant in the expression, it is impractical to actually use this to calculate inverses.
This general form also explains why the determinant must be nonzero for invertibility; as we are dividing through by its value.
An invertible matrix is also said to be nonsingular.
2 Calculating the Inverse By Hand
Method 1:
An easy way to calculate the inverse of a matrix by hand is to form an augmented matrix $[AI]$ from $A$ and $I_{n}$, then use Gaussian elimination to transform the left half into $I$. At the end of this procedure, the right half of the augmented matrix will be $A^{{1}}$ (that is, you will be left with $[IA^{{1}}]$).
Method 2:
One can calculate the $i,j$th element of the inverse by using the general formula; i.e.
$A^{{1}}_{{ji}}=C_{{ij}}(A)/\det{A}$ 
where $C_{{ij}}(A)$ is the $i,j$th cofactor expansion of the matrix $A$.
2.1 2by2 case:
For the $2\times 2$ case, the general formula reduces to a memorable shortcut. For the $2\times 2$ matrix
$M=\begin{bmatrix}a&b\\ c&d\end{bmatrix}$ 
The inverse $M^{{1}}$ is always
$M^{{1}}=\left(\frac{1}{\det M}\right)\begin{bmatrix}d&b\\ c&a\end{bmatrix}$ 
where $\det M$ is simply $adbc$.
2.2 Remarks
Some caveats: computing the matrix inverse for illconditioned matrices is errorprone; special care must be taken and there are sometimes special algorithms to calculate the inverse of certain classes of matrices (for example, Hilbert matrices).
3 Avoiding the Inverse and Numerical Calculation
The need to find the matrix inverse depends on the situation– whether done by hand or by computer, and whether the matrix is simply a part of some equation or expression or not.
Instead of computing the matrix $A^{{1}}$ as part of an equation or expression, it is nearly always better to use a matrix factorization instead. For example, when solving the system $Ax=b$, actually calculating $A^{{1}}$ to get $x=A^{{1}}b$ is discouraged. LUfactorization is typically used instead.
We can even use this fact to speed up our calculation of the inverse by itself. We can cast the problem as finding $X$ in
$AX=B$ 
For $n\times n$ matrices $A$, $X$, and $B$ (where $X=A^{{1}}$ and $B=I_{n}$). To solve this, we first find the $LU$ decomposition of $A$, then iterate over the columns, solving $Ly=Pb_{k}$ and $Ux_{k}=y$ each time ($k=1\ldots n$). The resulting values for $x_{k}$ are then the columns of $A^{{1}}$.
4 Elements of Invertible Matrices
Typically the matrix elements are members of a field when we are speaking of inverses (i.e. the reals, the complex numbers). However, the matrix inverse may exist in the case of the elements being members of a commutative ring, provided that the determinant of the matrix is a unit in the ring.
References
 1 Golub and Van Loan, “Matrix Computations,” Johns Hopkins Univ. Press, 1996.
 2 “Matrix Math,” http://easyweb.easynet.co.uk/ mrmeanie/matrix/matrices.htm
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Missing cached output! Please contact an admin
Why do we keep getting that {{Missing cached output! Please contact an admin}} error so often? I punch the <reload> button several times, all to âˆ… avail.
Jon
Re: Missing cached output! Please contact an admin
I got the same error twice some days ago while editing an entry.
At first I thought it was my fault for doing something wrong, so I just deleted the entry and entered it back from a backup on my computer.
But now I start thinking there could be a problem of some sort with the server and/or database...
Re: Missing cached output! Please contact an admin
RE:"I got the same error twice some days ago while editing an entry."
Here I was referring to an entry by someone else, which I cannot edit or rerender, but only hit the reload button. Still, I do often get this message on my own entries, and even then the reload and rerender do not always work, and so I have to go back and do a nullchange edit, which leads to an excess of apparent versions.
Jon
Re: Missing cached output! Please contact an admin
Jon Awbrey writes:
> Here I was referring to an entry by someone
> else, which I cannot edit or rerender
You can rerender any entry: copy the URL
http://planetmath.org/?op=rerender&from=objects&id=XXXX
into your browser's address bar, replace XXXX with the object's ID number, and hit Enter.
Re: Missing cached output! Please contact an admin
 You can rerender any entry: copy the URL

 http://planetmath.org/?op=rerender&from=objects&id=XXXX

 into your browser's address bar, replace XXXX with
 the object's ID number, and hit Enter.
Thanks, I can never seem to remember that one 
is it in a FAQ somewhere that I could bookmark?
At any rate, the re*render doesn't always work either,
even on my own entries, and so I have to do a nulledit.
Jon
Re: Missing cached output! Please contact an admin
Jon Awbrey wrote:
> Thanks, I can never seem to remember that one 
> is it in a FAQ somewhere that I could bookmark?
I don't think so. But you could click on the link
http://planetmath.org/?op=rerender&from=objects&id=XXXX
(which gives an error message) and bookmark that  that makes it quite convenient to use.
Re: Missing cached output! Please contact an admin
We are going to look into fixing this bug as part of the summer programming work.
apk