Issues with calculating the determinant of a matrix
I am trying to calculate the determinant of the inverse of a matrix. The inverse of the matrix exists. However, when I try to calculate the determinant of the inverse, it gives me Inf value in matlab. What is the reason behind this?
Short answer: given A = inv(B), then det(A)==Inf may have two explanations:
- an overflow during the numerical computation of the determinant,
- one or more infinite elements in A.
In the first case your matrix is badly scaled so that det(B) may underflow and det(A) overflow. Remember that det(a*B) == a^N * det(B) where a is a scalar and B is a N times N matrix.
In the second case (i.e. nnz(A==inf)>0) matrix B may be "singular to working precision".
A matrix is nearly singular if it has a large condition number. (A small determinant has nothing to do with singularity, since the magnitude of the determinant itself is affected by scaling.).
A matrix is singular to working precision if it has a zero pivot in the Gaussian elimination: when computing the inverse, matlab has to calculate 1/0 which returns Inf.
In fact in Matlab overflow and zero-division exceptions are not caught, so that, according to IEEE 754, an Inf value is propagated.