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Given the matrix:

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\$$A=\\begin{pmatrix} \\var{a11}& \\var{a12}\\\\ \\var{a21}&\\var{a22}\\end{pmatrix}\$$

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Evaluate the following expression:

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\$$\\left(A^2+\\var{k1}A+\\var{k2}I\\right)^T\$$ = []

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This question tests students knowledge of basic matrix arithmetic.

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\$$A=\\begin{pmatrix} \\var{a11}& \\var{a12}\\\\ \\var{a21}&\\var{a22}\\end{pmatrix}\$$

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\$$A^2=\\begin{pmatrix} \\var{a11}& \\var{a12}\\\\ \\var{a21}&\\var{a22}\\end{pmatrix}\\begin{pmatrix} \\var{a11}& \\var{a12}\\\\ \\var{a21}&\\var{a22}\\end{pmatrix}\$$

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Remember multiplication of matrices is carried out by multiplying the rows of the first matrix by the columns of the second matrix.

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\$$A^2=\\begin{pmatrix} \\var{a11}& \\var{a12}\\\\ \\var{a21}&\\var{a22}\\end{pmatrix}\\begin{pmatrix} \\var{a11}& \\var{a12}\\\\ \\var{a21}&\\var{a22}\\end{pmatrix}=\\begin{pmatrix}\\var{a11}*\\var{a11}+\\var{a12}*\\var{a21}&\\var{a11}*\\var{a12}+\\var{a12}*\\var{a22}\\\\ \\var{a21}*\\var{a11}+\\var{a22}*\\var{a21}&\\var{a21}*\\var{a12}+\\var{a22}*\\var{a22}\\end{pmatrix}\$$

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\$$A^2=\\begin{pmatrix}\\simplify{{a11}*{a11}+{a12}*{a21}}&\\simplify{{a11}*{a12}+{a12}*{a22}}\\\\ \\simplify{{a21}*{a11}+{a22}*{a21}}&\\simplify{{a21}*{a12}+{a22}*{a22}}\\end{pmatrix}\$$

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\$$\\var{k1}A=\\begin{pmatrix} \\var{k1}*\\var{a11}& \\var{k1}*\\var{a12}\\\\ \\var{k1}*\\var{a21}&\\var{k1}*\\var{a22}\\end{pmatrix}=\\begin{pmatrix} \\simplify{{k1}*{a11}}& \\simplify{{k1}*{a12}}\\\\ \\simplify{{k1}*{a21}}&\\simplify{{k1}*{a22}}\\end{pmatrix}\$$

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\$$\\left(A^2+\\var{k1}A+\\var{k2}I\\right)^t=\\left(\\begin{pmatrix}\\simplify{{a11}*{a11}+{a12}*{a21}}&\\simplify{{a11}*{a12}+{a12}*{a22}}\\\\ \\simplify{{a21}*{a11}+{a22}*{a21}}&\\simplify{{a21}*{a12}+{a22}*{a22}}\\end{pmatrix}+\\begin{pmatrix} \\simplify{{k1}*{a11}}& \\simplify{{k1}*{a12}}\\\\ \\simplify{{k1}*{a21}}&\\simplify{{k1}*{a22}}\\end{pmatrix}+\\begin{pmatrix} \\var{k2}&0\\\\0&\\var{k2}\\end{pmatrix}\\right)^t\$$

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\$$\\left(A^2+\\var{k1}A+\\var{k2}I\\right)^t=\\begin{pmatrix}\\simplify{{a11}*{a11}+{a12}*{a21}+{k1}{a11}+{k2}}&\\simplify{{a11}*{a12}+{a12}*{a22}+{k1}*{a12}}\\\\ \\simplify{{a21}*{a11}+{a22}*{a21}+{k1}*{a21}}&\\simplify{{a21}*{a12}+{a22}*{a22}+{k1}*{a22}+{k2}}\\end{pmatrix}^t\$$

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\$$\\left(A^2+\\var{k1}A+\\var{k2}I\\right)^t=\\begin{pmatrix}\\simplify{{a11}*{a11}+{a12}*{a21}+{k1}{a11}+{k2}}&\\simplify{{a21}*{a11}+{a22}*{a21}+{k1}*{a21}}\\\\ \\simplify{{a11}*{a12}+{a12}*{a22}+{k1}*{a12}}&\\simplify{{a21}*{a12}+{a22}*{a22}+{k1}*{a22}+{k2}}\\end{pmatrix}\$$

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