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

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)

Thanks in advance

[James Tursa]

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]

[ -(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.

Answer 1

[JESUS DAVID ARIZA ROYETH]

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.