⎡a22a33−a23a32a23a31−a21a33a21a32−a22a31a13a32−a12a33a11a33−a13a31a12a31−a11a32a12a23−a13a22a13a21−a11a23a11a22−a12a21⎦⎤ as for 2x2 and 3x3 matrices.
aa−1=I
a:a−T=3
[ab]−1=b−1a−1
[ka]−1=k1a−1
[a+b]−1 has no simple formula
∂a∂a−1=−a−1⊗a−T Use the Voigt matrix representation and linear algebra rules. However, some special cases exist, e.g.
A: A−1= I
[ A: B]−1= B−1: A−1
[k A]−1=k1 A−1
[a⊗b]−1=a−1⊗b−1