Matrix solution of linear systems

[Tempest]

When r is less than n .

I'm curious if there's a way to do this.

Basically I need to find a way to solve a linear system where r is less than n with a matrix solution that's not the Gauss method, but the matrix equation method.

Now, if I'm right about the upper equation being used for this method, what follows after it ? Do you guys know of a possible matrix solution of linear systems that starts out like that ?

I found one way in a book, it's basically used when det A == 0 . It's a formula that goes like this:

The A and X in the above formula have the character ~ above them.

 

[mmm4444bot]

Now, if I'm right about the upper equation being used for this method, what follows after it ?

The next line.

A[sup:34paqpr0]-1[/sup:34paqpr0] is the notation for the inverse matrix of A.

If you multiply a matrix by its inverse, you get 1.

A X = B

A[sup:34paqpr0]-1[/sup:34paqpr0] * A * X = A[sup:34paqpr0]-1[/sup:34paqpr0] * B

1 * X = A[sup:34paqpr0]-1[/sup:34paqpr0] * B

X = A[sup:34paqpr0]-1[/sup:34paqpr0] * B

So, solving for matrix X requires multiplying matrix B by the inverse of matrix A.

Note: All asterisks signify matrix multiplication.

 

[Tempest]

Yes, I know that, but there's a 'special' way to do it when R (rank of the matrix) is less than N (number of unknowns). One way is that other formula I used, but is there anything else ?