Inverse of a matrix

[aaram81]

I need a help in C++ for the following question:
write a p[rogram to read 4 by 4 matrix, store it and find the inverse of this matrix. then multiply the inverse with the original matrix , store it . lastly print the inverse of the matrix which is output of multiplication.

Could you help me please...somebody..????

[mattmao]

Hello aaram81:

Conceputally speaking, you can do this program in this way:

create a two-dimensional array, array[4][4], each of the "cell" -- array[x][y] would be used to store its corresponding value in the matrix.

Create another two-dimensional array inversed[4][4] for storing the inversed array.

Let the user key in the values and store them in the array[4][4]

Then use for loops to iterate through all the cells of the array to inverse it, say,
inversed[x][y] = array[y][x];

There you go, you got the inversed array, and you can do more from that basis.


My logic is fine, the only thing that lacks here is the implementation of coding, you can do that yourself:)


Sincerely yours,

Matt

[cellerit]

mattmao i think you are confusing matrix inversion with matrix transposition.
see: http://en.wikipedia.org/wiki/Inverse_matrix

Inverting a matrix is a bit more complex. One solution would be to calculate its determinant and use:
B being the inversed matrix of A:
B = (1/determinant(A)) * Ck(A)

Please note that if determinant(A) = 0, then the matrix cannot be inversed, therefore you should stop the process at this point.

Ck(A) is the matrix of cofactors of A. To calculate the Ck(A) matrix you have to build a matrix of Minors of A then apply the right sign to each cell.
see: http://en.wikipedia.org/wiki/Cofactor_%28mathematics%29

Minor(A,i,j) would be the determinant of A when removing line i and column j.
So here you have to calculate a determinant for each cell, which makes 16 determinants. You then have the matrix of minors, lets call it M.

To obtain Ck(A) you simply change the sign of the cells of M in the following manner: Ck(A)[i][j] = (-1)^(i+j)*M[i][j]

You now have Ck(A) and can get B using the following formula:
B = (1/determinant(A)) * Ck(A)

[JosAH]

Never explicitly calculate the inverse of a matrix; it's numerically unstable. Use
a LUP decomposition instead: let L and U be lower and upper triangular matrixes
and let P be a row permutation matrix such that LU = PA. A permutation matrix
has exactly one 1 per row and column (all the other elements are zero).

Finding matrixes L and U is just a repeated Gauss row operation and P takes
care that the largest absolute pivot element is used.

For a linear system Ax= b:

PAx= Pb (permute the elements of vector b) -->
(Pb= b') LUx= b' -->
Ly = b' (find y by backward substitution) -->
Ux = y (find x by forward substitution).

Google is your friend: LUP decomposition.

kind regards,

Jos