Inverse of square matrix

wintersolstice · 2010-07-11 06:50:13

If you take the square matrix:

It's inverse is

Doesn't sound very interesting!

What about this?

consider this matrix

Now if

means the rth row and cth column element.

now delete the rth row and cth column to make a n-1 X n-1 matrix find it's determinent and call it

in the below matrix

is represented by

form the matrix

(the plus/minus sign is becuase you need to know the size of the matrix to know the sign of the bottom row)

In words what has been done:

change lowercase to capital (or from E to D)

start with the outer two elements which form diagonals swap the elements round (there's only one element in each so nothing changes)

go to the next diagonal pair (one inwards on either side) and swap them (the elements within the diagonal see above) and keep going

also the sign of the outermost diagonals stay the same the next pair change sign, whether the trailing diagonal (what all these diagonals are parallel to) changes depends on the size.

Calculate the determinent Δ in the following way:

(multiply the first row of original matrix by the first column of the new matrix)

To find the detiminent of the n-1 X n-1 matrices use the above method (which means looking at n-2 X n-2 matrices and so on down to 2 X 2 whose determinent is above

if you put say that the determinent of a 1 X 1 matrix is its one element (which means the method above dosen't work on 1 X 1)

then for the 2 X 2:

then right:

and sustitute

which fits the above

using this method you can find the inverse of a n X n matrix provided you know how to do the n-1 X n-1 (you just have to work your way backwards to the 2 X 2

unfortunatly I don't know how to prove it!

George,Y · 2010-07-25 14:58:51

Determinant is created to solve equations using the Crammer's Rule.
So you need to be familiar with properties of such a determinant first,
then can you go ahead to understand how it works in solving a group of linear equations via Crammer's Rule
It is really a smart mathematic invention!
So far!

wintersolstice · 2010-07-26 00:03:31

George,Y wrote:

Determinant is created to solve equations using the Crammer's Rule.
So you need to be familiar with properties of such a determinant first,
then can you go ahead to understand how it works in solving a group of linear equations via Crammer's Rule
It is really a smart mathematic invention!
So far!

I don't understand what you mean. I don't know if I've actually achieved anything yet

George,Y · 2010-07-26 18:27:05

sorry for the typo Cramer mistaken as Crammer

Here is what the determinants are created for

click here

Math Is Fun Forum

#1 2010-07-11 06:50:13

Inverse of square matrix

#2 2010-07-25 14:58:51

Re: Inverse of square matrix

#3 2010-07-26 00:03:31

Re: Inverse of square matrix

#4 2010-07-26 18:27:05

Re: Inverse of square matrix

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