Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2023 / Paper 2 / 6 / c

Write the inverse tensor asInsert this into $({\delta }_{ik}+\lambda X_{ik})Y_{kj}={\delta }_{ij}$ and equate powers of $\lambda$. One obtains successivelyThus the Neumann series givesEvery contraction is an ordinary ordered matrix product. No transpose or symmetry assumption is used. Multiplying the truncated series on either side by $I+\lambda X$ leaves $I-{\lambda }^4{X}^4$, which verifies the error order. For a fixed finite matrix this expansion converges whenever $|\lambda |\Vert X\Vert <1$ in a submultiplicative matrix norm.