# Checking invertiblity of a symbolic matrix (small size N=12)

3 views (last 30 days)
SandeepKumar R on 30 Jul 2020
Edited: SandeepKumar R on 1 Aug 2020
Hello everyone,
I have a symbolic matrix A of size 12x12. I need to check if inverse of this matrix exists. If rank(A)=12, then I know that I can invert it. However, as explained in the official documentation of mathworks, this is not always correct. Another approach is to compute row reduced echelon form of A i.e. rref(A) and check if it is rref(A)=I (identity matrix) to ensure invertibility. If I use a symbolic A and found rref(A)=I,can the invertibilty be guaranteed?
(I can evaluate the matrix A numerically and check invertibility however I don't want to do it for possible cases)

James Tursa on 30 Jul 2020
The inverse of a generic 12x12 symbolic matrix is going to have a gazillion terms that will be intractable to analyze. Does your matrix have a special form? What do the terms of A look like?
SandeepKumar R on 1 Aug 2020
[ -(1.6261e-9*Y(a,b,c,d))/v, 0, -(1.78e-10*Y(a,b,c,d))/v, 0, 0.28971, 0.28971*b]
[ 0, 0.037952, 0, -0.11328, -0.28971*b, 0.28971]
[ -1.0163, 0, -0.11125, 0, -0.28971*a, 0]
[ 0, -0.60057, 0, -0.018252, 0, 0.033317*a]
[ 0.050267, 0, -0.10264, 0, 0, -0.033317*a*b]
[ 0, -0.039111, 0, 0.10082, 0, 0.033317]
Consider the 6x6 matrix for brevity. The matrix looks like this after simplification. Y is function of a,b c,d. I want check if system is invertible rather than computing inv(A). rref(A) gives an identity matrix and this implies that A is invertible. However, I want to know if there are any limitations of rref() on symbolic matrices like rank (link) .
For many cases tried by substituting and evaluating the A numerically, the matrix was invertible.

JESUS DAVID ARIZA ROYETH on 30 Jul 2020
remember that a matrix has an inverse if and only if its determinant is different from 0, therefore you must calculate for which conditions the determinant of A "det(A)" is different from 0, if it is true for whatever the value of your variables, then that symbolic matrix will always be invertible.