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2014 CHMMC Tiebreaker 2 - square routs od 2x2 matrix

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March 1, 2024
matrixMatriceslinear algebraCHMMC

Problem Statement

A matrix [xyzw]\begin{bmatrix} x & y \\ z & w \end{bmatrix} has square root [abcd]\begin{bmatrix} a & b \\ c & d \end{bmatrix} if [abcd]2=[a2+bcab+bdac+cdbc+d2]=[xyzw]\begin{bmatrix} a & b \\ c & d \end{bmatrix}^2 = \begin{bmatrix} a^2 + bc &ab + bd \\ ac + cd & bc + d^2 \end{bmatrix} = \begin{bmatrix} x & y \\ z & w \end{bmatrix} Determine how many square roots the matrix [2234]\begin{bmatrix} 2 & 2 \\ 3 & 4 \end{bmatrix} has (complex coefficients are allowed).




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