Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2001 / Paper 1 / 10

Use the Einstein summation convention, with repeated component indices summed. The following three expressions are listed separately below. For cyclic invariance of the trace,where the middle equality commutes scalar entries, not the matrices themselves. If $A$ is invertible, this givesThis is similarity invariance of trace.

Taking the diagonal sum of the matrix product givesBoth repeated indices are summed from one to $n$.

Apply row-column matrix multiplication twice:The indices $i,j$ are free, while $k,\ell$ are summed.

The identity matrix has entries given by the Kronecker delta, soEach of the $n$ diagonal entries is one.

Expand $(I+B{)}^2$ in components:If $B^2=0$, all powers $B^k$ for $k\ge 2$ vanish. The matrix binomial theorem, valid here because $I$ commutes with $B$, therefore yieldsThese powers of the identity plus a square-zero matrix can also be proved by induction: $(I+nB)(I+B)=I+(n+1)B$. For the displayed example, $B^2=0$ directly, andThis is the repeated function composition of a shear mapping. The argument gives every nonnegative integer $n$; since $(I+B{)}^{-1}=I-B$, the same formula also holds for negative integers.