how do you invert a 3x3 matrix?

Answer 1

There are actually 3 good method for doing this.
I'll let you try them out on the matrix
A =
2 -1 1
-1 2 1
1 -1 2

1). The Gauss, Jordan method mentioned in a
previous post.
You write down A and put the 3x3 identity
matrix, I, to its right.
Then you use elementary row operations
on the whole 3x6 system to reduce A to the
identity matrix. The right-hand 3x3 matrix
then gives the inverse of A.
Of course, if you run into a row of zeros
at any point, A is singular and has no inverse.
Try it for A given above. The calculations
are too messy and too miserable to write here!

2). Another method for computing the inverse
of A is to use the result
A^-1 = adj[A] / det A.
Here adj[A] is the adjoint matrix of A.
You get it by replacing every element of A by
its signed minor, then taking the transpose
of the resulting matrix.
For example,
the determinant of A(given above) is 6
and adj[A] is
5 1 -3
3 3 -3
-1 1 3
Again, I'll let you verify the details.

3). Faddeev's method.
I like this method because you not
only pick up the determinant of A
and the coefficients of its characteristic polynomial,
but it also generalises to an nxn matrix very nicely,
whereas the other 2 methods are sooo messy for
large matrices.
An excellent description of the method is
given on the website
math.fullerton.edu/mathews/n2003/FaddeevLeverrierMod
.html
If you run into p_n = 0, that means A is singular
and has no inverse.
Let me give you the results for A(given above).
I'll just use numbers instead of subscripts here.
A1 = A = 2 -1 1
................-1 2 1 ........ p1 = tr(A) = 6 B1 = A - p1*I
..................1 1 -2

B1 = -4 -1 1
.........-1 -4 1
...........1 -1 -4

A2 = AB1 = -6 1 -3
.......................3. 8 -3 ........p2 = ½(-6 -8 -8) = -11
.......................-1 1 -8

B2 = A2 -p2I =
5 1 -3
3 3 -3
-1 1 3
Note that we have found adj(A) here.
B_(n-1) always gives adj(A). So the next matrix
should be (det A)*I, and, in fact,
A3 = AB2 =
6 0 0
0 6 0
0 0 6
and p3 = 6 = det(A).
Hope that helps!

Answer 2

An easier way to invert a 3x3 matrix is to set it equal to the identity matrix (on the right side) and use elementary row operations until you have the identity matrix on the left side instead. The above mentioned website has a great formula that works really well on 2x2 matrices but is rather tedious when you apply it to a 3x3 matrix.

Answer 3

The pentagram will be inverted with one element down. The implication is of spirit subservient to count number, of guy subservient to his carnal needs. The inverted pentagram has end up seen with the help of many pagans as representing the darkish area and that is abhored as an evil image. wide-spread christians, certainly, see any type of pentagram as such. although, those are modern-day advancements and the inverted pentagram is the variety of Gardnerian 2d degree initiation, representing the will of the witch to detect undemanding procedures to face the darkness interior so as that it would want to no longer later upward push as a lot as take administration. The centre of a pentagram implies a sixth formative aspect - love/will which controls from interior, ruling count number and spirit with the help of Will and the managed magickal course of sexual energies. it really is yet another lesson of initiation.

Answer 4

You can find a method on this website:
http://mathworld.wolfram.com/MatrixInverse.html

It will be quite a mess, to type that out here and now.