IMC 2004 · Day 2 · P7

Let $A = \begin{pmatrix} A_1 \\ A_2 \end{pmatrix}$ and $B = \begin{pmatrix} B_1 & B_2 \end{pmatrix}$ where $A_1, A_2, B_1, B_2$ are $2 \times 2$ matrices. Then $$\begin{pmatrix} 1 & 0 & -1 & 0 \\ 0 & 1 & 0 & -1 \\ -1 & 0 & 1 & 0 \\ 0 & -1 & 0 & 1 \end{pmatrix} = \begin{pmatrix} A_1 \\ A_2 \end{pmatrix} \begin{pmatrix} B_1 & B_2 \end{pmatrix} = \begin{pmatrix} A_1 B_1 & A_1 B_2 \\ A_2 B_1 & A_2 B_2 \end{pmatrix}$$ therefore, $A_1 B_1 = A_2 B_2 = I_2$ and $A_1 B_2 = A_2 B_1 = -I_2$. Then $B_1 = A_1^{-1}$, $B_2 = -A_1^{-1}$ and $A_2 = B_2^{-1} = -A_1$. Finally, $$BA = \begin{pmatrix} B_1 & B_2 \end{pmatrix} \begin{pmatrix} A_1 \\ A_2 \end{pmatrix} = B_1 A_1 + B_2 A_2 = 2 I_2 = \begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix}$$