This gives the system of equations:

February 22, 2026

This gives the system of equations:

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5**Question:** A linguist studying languages is interested in the symmetry of phonetic transformations. Consider a transformation matrix \( T \) such that \( T^2 = I \), where \( I \) is the identity matrix. If \( T = egin{pmatrix} a & b \ c & d \end{pmatrix} \), find the conditions on \( a, b, c, \) and \( d \) for \( T \) to be a valid symmetry transformation.
Solution:** For \( T \) to satisfy \( T^2 = I \), we have:
egin{pmatrix} a & b \ c & d \end{pmatrix} egin{pmatrix} a & b \ c & d \end{pmatrix} = egin{pmatrix} a^2 + bc & ab + bd \ ac + cd & bc + d^2 \end{pmatrix} = egin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix}
\( a^2 + bc = 1 \)
\( ab + bd = 0 \)
\( ac + cd = 0 \)

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