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Home » Thus, the length is \(2 \times 40 = 80\). - AMAZONAWS

Thus, the length is \(2 \times 40 = 80\). - AMAZONAWS

📅 March 5, 2026 👤 scraface

Mar 05, 2026

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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}

📌 This gives the system of equations:

📌 \( a^2 + bc = 1 \)

📌 \( ab + bd = 0 \)

📌 \( ac + cd = 0 \)

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