Inverse

2nd order tensors

Calculation

The inverse of 2nd order tensors a\boldsymbol{ a} in 2d and 3d are

2d: [aij]−1=1det(a)[a22−a12−a21a11]3d: [aij]−1=1det(a)[a22a33−a23a32a13a32−a12a33a12a23−a13a22a23a31−a21a33a11a33−a13a31a13a21−a11a23a21a32−a22a31a12a31−a11a32a11a22−a12a21]\begin{aligned} \text{2d: }[a_{ij}]^{-1} &= \frac{1}{\mathrm{det}(\boldsymbol{ a})} \begin{bmatrix} a_{22} & -a_{12} \\ -a_{21} & a_{11} \end{bmatrix} \\ \text{3d: }[a_{ij}]^{-1} &= \frac{1}{\mathrm{det}(\boldsymbol{ a})} \begin{bmatrix} a_{22} a_{33} - a_{23} a_{32} & a_{13} a_{32} - a_{12} a_{33} & a_{12} a_{23} - a_{13} a_{22} \\ a_{23} a_{31} - a_{21} a_{33} & a_{11} a_{33} - a_{13} a_{31} & a_{13} a_{21} - a_{11} a_{23} \\ a_{21} a_{32} - a_{22} a_{31} & a_{12} a_{31} - a_{11} a_{32} & a_{11} a_{22} - a_{12} a_{21} \end{bmatrix} \end{aligned}

as for 2x2 and 3x3 matrices.

Properties

  • aa−1=I\boldsymbol{ a}\boldsymbol{ a}^{-1} = \boldsymbol{ I}

  • a:a−T=3\boldsymbol{ a}:\boldsymbol{ a}^{-\mathrm{T}} = 3

  • [ab]−1=b−1a−1\left[\boldsymbol{ a}\boldsymbol{ b}\right]^{-1} = \boldsymbol{ b}^{-1}\boldsymbol{ a}^{-1}

  • [ka]−1=1ka−1\left[k\boldsymbol{ a}\right]^{-1} = \frac{1}{k}\boldsymbol{ a}^{-1}

  • [a+b]−1\left[\boldsymbol{ a} + \boldsymbol{ b}\right]^{-1}\quad has no simple formula

Differentiation

∂a−1∂a=−a−1⊗‾a−T\begin{aligned} \frac{\partial \boldsymbol{ a}^{-1}}{\partial \boldsymbol{ a}} = -\boldsymbol{ a}^{-1}\overline{\otimes}\boldsymbol{ a}^{-\mathrm{T}} \end{aligned}

4th order tensors

Use the Voigt matrix representation and linear algebra rules. However, some special cases exist, e.g.

  • A: A−1= I\textbf{\textsf{ A}}:\textbf{\textsf{ A}}^{-1}=\textbf{\textsf{ I}}

  • [ A: B]−1= B−1: A−1\left[\textbf{\textsf{ A}}:\textbf{\textsf{ B}}\right]^{-1} = \textbf{\textsf{ B}}^{-1}:\textbf{\textsf{ A}}^{-1}

  • [k A]−1=1k A−1\left[k\textbf{\textsf{ A}}\right]^{-1} = \frac{1}{k}\textbf{\textsf{ A}}^{-1}

  • [a⊗‾b]−1=a−1⊗‾b−1\left[\boldsymbol{ a}\overline{\otimes}\boldsymbol{ b}\right]^{-1} = \boldsymbol{ a}^{-1}\overline{\otimes}\boldsymbol{ b}^{-1}

© Knut Andreas Meyer.
Last modified: October 31, 2025.
Website built with Franklin.jl and Julia.