Matrices

[deadly7]

So I was sitting around in English and I'm wondering "How do Matrix Inverses work?"  I tried to find a pattern but I couldn't.
Help?

[dark_drake]

I learned how to do these long ago.  The only ones I remember now are the 2x2 matrices.

The matrix is:             | a  b |
                               | c  d |

The inverse is:      1       | d   -b |
                       ad-bc   | -c   a |

[Rule]

I don't think there is a formula for getting the inverse of > nxn matrices, where n = 2.

One way to find the inverse of larger square matrices is to figure out what operations you would have to do to turn the matrix (A) into the identity matrix.  For example, 1 step might be to multiply the first row by 4. 
The matrix [4  |   4   |  4 | ... |  4]  would accomplish this.  Let that matrix be called B.  Then you
                  [ 0........................
                    .........................     ]

may need to subtract the first row from the second.  Let the matrix that accomplishes this be called C. 
Let's say then that,

E.g.       C*B*A = I.     By definition, the matrix, G = C*B is the inverse of
A.

[dark_drake]

I don't think there is a formula for getting the inverse of > nxn matrices, where n = 2.

There is, but I forgot it.

[Rule]

Ah, there's a canned formula for 3x3.

For,


The inverse is :

(|A| = det(A) )

For higher dimensional matrices no-one has bothered with a formula.  I think even for 3x3 matrices it's easier just using the method I described in my previous post (rather than the formula).

"A general nxn matrix can be inverted using methods such as the Gauss-Jordan elimination, Gaussian elimination, or LU decomposition."  -- Mathworld.

For more information, see:
http://mathworld.wolfram.com/MatrixInverse.html


Edit:
My mistake...
 In general, the inverse of an  N x N matrix A is

                                 
    A^-1 =  [1/det(A)]* adj(A)
                               

[d&q]

Since you're in Algebra 2, I think all you would need to know is this:

The matrix is:             | a  b |
                                 | c  d |

The inverse is:      1       | d   -b |
                         ad-bc   | -c   a |

 

Btw, 1/ad-bc is 1/(det), if you didn't already know that.

And didn't I show you this before?



I wonder what's on TV tonight...

[deadly7]

No.  We never had to know how to manually do inverses, but I really wanted to know.
Thanks a ton.

[Chavo]

I was wondering there for a minute...I didn't cover this stuff until Calc 3

[dark_drake]

We covered 2x2 and 3x3 matrices and their inverses in Algebra 2. 

[Chavo]

Maybe I just don't remember covering it in Algebra 2, I certainly didn't do anything with matrices from Trig through Calc 2

[dark_drake]

You probably did cover it Algebra 2, but once your through the section in Algebra 2, you don't see it again for years.  I didn't see it in Pre-Calc or my AP Calc class.

[Chavo]

Probably so, Algebra 2 was ....6 years ago for me.

[d&q]

They cover it in our Pre-Calc class, and very briefly(for review) in Calc.

[Rule]

They cover it in our Pre-Calc class, and very briefly(for review) in Calc.

Linear algebra becomes extremely important in Advanced Calculus; however, whether or not you know it should have little or no effect on how well you do in first year calc or AP Calc (AB/BC).

Actually, I only ever took a formal class in linear algebra in 2nd year (before that I only had a vague idea of what a matrix was).   Although, it was a very comprehensive course.  That's what my university was like -- completely avoid talking about a concept (e.g. imaginary numbers), and then, (at an arbitrary time), tell you more than most students would ever care to know about it.  My education was very discretized (unlike Calculus) .