역행렬 곱 성분 결정

$\displaystyle A ^{- 1} B ^{- 1} = {\begin{pmatrix} - 1 & & b \\ 0 & & 1 \end{pmatrix}}$에서 $\displaystyle A ^{- 1} B ^{- 1} = ( BA ) ^{- 1} = {\begin{pmatrix} - 1 & & b \\ 0 & & 1 \end{pmatrix}}$ $\displaystyle \therefore BA = {\begin{pmatrix} - 1 & & b \\ 0 & & 1 \end{pmatrix}} ^{- 1}$$\displaystyle = {\begin{pmatrix} - 1 & & b \\ 0 & & 1 \end{pmatrix}}$ 이 때, $\displaystyle BA = {\begin{pmatrix} 2 & & - 3 \\ a & & - 1 \end{pmatrix}} {\begin{pmatrix} 1 & & 2 \\ 1 & & 1 \end{pmatrix}} = {\begin{pmatrix} - 1 & & 1 \\ a - 1 & & 2 a - 1 \end{pmatrix}}$이므로 $\displaystyle {\begin{pmatrix} - 1 & & 1 \\ a - 1 & & 2 a - 1 \end{pmatrix}} = {\begin{pmatrix} - 1 & & b \\ 0 & & 1 \end{pmatrix}}$ $\therefore a = 1 , b = 1$ $\therefore a + b = 2$