# Paper 1, Section II, A

Let $A$ and $B$ be real $n \times n$ matrices.

(i) Define the trace of $A, \operatorname{tr}(A)$, and show that $\operatorname{tr}\left(A^{T} B\right)=\operatorname{tr}\left(B^{T} A\right)$.

(ii) Show that $\operatorname{tr}\left(A^{T} A\right) \geqslant 0$, with $\operatorname{tr}\left(A^{T} A\right)=0$ if and only if $A$ is the zero matrix. Hence show that

$\left(\operatorname{tr}\left(A^{T} B\right)\right)^{2} \leqslant \operatorname{tr}\left(A^{T} A\right) \operatorname{tr}\left(B^{T} B\right)$

Under what condition on $A$ and $B$ is equality achieved?

(iii) Find a basis for the subspace of $2 \times 2$ matrices $X$ such that

$\begin{gathered} \operatorname{tr}\left(A^{T} X\right)=\operatorname{tr}\left(B^{T} X\right)=\operatorname{tr}\left(C^{T} X\right)=0 \\ \text { where } \quad A=\left(\begin{array}{ll} 1 & 1 \\ 2 & 0 \end{array}\right), \quad B=\left(\begin{array}{rr} 1 & 1 \\ 0 & -2 \end{array}\right), \quad C=\left(\begin{array}{ll} 0 & 0 \\ 1 & 1 \end{array}\right) \end{gathered}$