Is it possible to invert an n*n matrix (square matrix) with every (i.e. n) entry down the main diagonal = 0?

Question

I need to invert an n*n matrix where the values down the main diagonal are all equal to zero. I've tried usingh numerical recipes such a Lower/Upper decomposition and Singular Value Decomposition, yet these seem to give out an error message. Is it actually possible to calculate the inverse of such a matrix?

Answer 1

It depends on the matrix. There would have to be other non-zero values off the diagonal. In that case, use elementary row operations of the adjunct matrix
[A|I]
until you get a matrix of the form
[I|B].
Then B will be your inverse. If you cannot get I on the left hand side using ERO's then A will not be invertible.

Answer 2

each and every component to that matrix must be both 0 or a million........ so the placement (a million,a million) must be filled with the help of two approaches. it truly is real for all positions so kind of such matrices are 2^9 as there are 9 positions in a 3x3 matrix. complete kind of such matrices are 512

Answer 3

Yes is it possible..
for Matrix A:

Inv(A) = Adj(A) / Det(A)

where :
Adj(A) = Adjoint matrix (A)
Det(A) = Determinant(A)
You have to check the determinant (Det(A)) if it is equal to zero then the matrix has no inverse. a matrix CAN have a Determinant not equal to zero value even with all the main diagonal entries as zeros.