Evaluate the following expression:
\n\\(\\left(A^2+\\var{k1}A+\\var{k2}I\\right)^T\\) = [[0]]
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", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "variable_groups": [], "extensions": [], "preamble": {"js": "", "css": ""}, "statement": "Given the matrix:
\n\\(A=\\begin{pmatrix} \\var{a11}& \\var{a12}\\\\ \\var{a21}&\\var{a22}\\end{pmatrix}\\)
\n", "variables": {"a12": {"templateType": "randrange", "name": "a12", "definition": "random(0 .. 10#1)", "group": "Ungrouped variables", "description": ""}, "a21": {"templateType": "randrange", "name": "a21", "definition": "random(2 .. 9#1)", "group": "Ungrouped variables", "description": ""}, "a11": {"templateType": "randrange", "name": "a11", "definition": "random(1 .. 10#1)", "group": "Ungrouped variables", "description": ""}, "b11": {"templateType": "anything", "name": "b11", "definition": "{a11}^2+{a12}*{a21}+{k1}*{a11}+{k2}", "group": "Ungrouped variables", "description": ""}, "a22": {"templateType": "randrange", "name": "a22", "definition": "random(11 .. 21#1)", "group": "Ungrouped variables", "description": ""}, "b21": {"templateType": "anything", "name": "b21", "definition": "{a12}*{a11}+{a22}*{a12}+{k1}*{a12}", "group": "Ungrouped variables", "description": ""}, "b22": {"templateType": "anything", "name": "b22", "definition": "{a21}*{a12}+{a22}^2+{k1}*{a22}+{k2}", "group": "Ungrouped variables", "description": ""}, "b12": {"templateType": "anything", "name": "b12", "definition": "{a21}*{a11}+{a22}*{a21}+{k1}*{a21}", "group": "Ungrouped variables", "description": ""}, "k2": {"templateType": "randrange", "name": "k2", "definition": "random(6 .. 12#1)", "group": "Ungrouped variables", "description": ""}, "k1": {"templateType": "randrange", "name": "k1", "definition": "random(2 .. 7#1)", "group": "Ungrouped variables", "description": ""}}, "advice": "\\(A=\\begin{pmatrix} \\var{a11}& \\var{a12}\\\\ \\var{a21}&\\var{a22}\\end{pmatrix}\\)
\n\\(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}\\)
\nRemember multiplication of matrices is carried out by multiplying the rows of the first matrix by the columns of the second matrix.
\n\\(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}\\)
\n\\(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}\\)
\n\\(\\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}\\)
\n\\(\\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\\)
\n\\(\\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\\)
\n\\(\\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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