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The matrices

\(\large{ \mathbf{A} = \begin{pmatrix} 1 &x \\ y & 1 \end{pmatrix} \text{ and } \mathbf{B} = \begin{pmatrix} -1 &x \\ y & -1 \end{pmatrix} }\)

are inverses.

What is \(xy\)?

\(\begin{array}{|lrcll|} \hline & AB&=&I \\ & \begin{pmatrix} 1 &x \\ y & 1 \end{pmatrix} \begin{pmatrix} -1 &x \\ y & -1 \end{pmatrix} &=& \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \\\\ & \begin{pmatrix} -1+xy &x-x \\ -y+y & xy-1 \end{pmatrix} &=& \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \\\\ & \begin{pmatrix} xy-1 &0 \\ 0 & xy-1 \end{pmatrix} &=& \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \\ \hline \end{array} \)

\(\begin{array}{|rcll|} \hline xy-1 &=& 1 \\ \mathbf{xy} & \mathbf{=} & \mathbf{2} \\ \hline \end{array}\)