Need help with the inverse of a 2X2 matrix!?

Question

How can matrix A =
[5 8]
[17 3]

have an inverse that is
[9 2]
[1 15]

I know the process of getting a 2X2 matrix, but my solution results with a lot of fractions, and also the calculator's.

This is in my math book, and provides no explanation as to how it is done. We are suppose to know =S.

And it is also an inverse since A times that also gives the identity matrix... But how can one get that solution straightforward (a solution with all integers)?

Thanks!
(This is college math)

everything is mod26.
so 5*9 + 8*1 = 51 but that mod26 gives 1. That is the first one in the identity matrix.

This is really confusing I know.

Answer 1

It can't. If you multiply them, you will get
[5 8] . [9 2]
[17 3] [1 15]
=
[53 130]
[156 79]

which is nothing like the identity matrix!
The correct inverse for A - if you want to check your results - is
[-3/121 8/121]
[17/121 -5/121]

[Edit: after additional information was given, in particular that all calculations are mod 26]
... Ah! It helps if you include highly important details like that...

Normally, for a 2x2 matrix
[a b]
[c d]
you evaluate the determinant ad-bc and divide the matrix
[d -b]
[-c a]
by it to get the inverse. You follow the same process for working mod 26, but the actual calculations are different.

OK, the determinant (ad - bc) is -121; working mod 26 this is equivalent to 9. Instead of dividing by 9, we multiply by its inverse, which modulo 26 is 3 (3 * 9 = 27 = 1 mod 26). So we get the inverse as
[3 -8] . 3
[-17 5]
= (mod 26)
[3 18] . 3
[9 5]
=
[9 54]
[27 15]
= (mod 26)
[9 2]
[1 15].

Answer 2

Damned if *I* know since the product of those two matrices is certainly not the identity matrix.

In general, the best way to get the inverse of a matrix 'A' is to replace each element (a(i,j)) of the matrix with its 'co-factor' (the value that you get by evaluating the determinant of the matrix with the i'th row and j'th column removed) multiplied by
(-1)^(i+j). This is called the 'adjoint' (or classical adjoint) of A. Now form the transpose of the adjoint by 'swaping'the rows and columns (that is, a(i,j) gets swapped with a(j,i) for each a in A). Finally, divide each element of the adjoint transpose by the determinant of the original matrix and you have the inverse.

In a 2X2 matrix
|a b|
|c d|
the determinant it ad-bc and the inverse is
|d/(ad-bc) -b/(ad-bc)|
|-c/(ad-bc) a/(ad-bc)|

The 2X2 case is not too bad, but for matrices of order 3 and up, the amount of computation and 'book-keeping' becomes a real pain in the αss. However..... there are a couple of sites that have some pretty good interactive calculators
http://wims.unice.fr/wims/wims.cgi?lang=en&+session=70C7B3F82D.1

HTH

Doug

Answer 3

The inverse of [4 -1] [2 -1] is [1/2 -1/2] [1 -2], not the matrix D.

Answer 4

The rule for inverting a 2x2 matrix:

[a b]
[c d]

is

1 / (ad-bc) times
[d -b]
[-c a]