جواب کار در کلاس صفحه 19 درس 1 هندسه دوازدهم (ماتریس و کاربردها)

\(\begin{array}{*{20}{l}}\begin{array}{l}A \times B = \\\\\left[ {\begin{array}{*{20}{c}}1&2\\{ - 1}&3\end{array}} \right] \times \left[ {\begin{array}{*{20}{c}}{ - 1}&{ - 4}\\1&0\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}1&{ - 4}\\4&4\end{array}} \right]\end{array}\\{}\\\begin{array}{l}B \times A = \\\\\left[ {\begin{array}{*{20}{c}}{ - 1}&{ - 4}\\1&0\end{array}} \right] \times \left[ {\begin{array}{*{20}{c}}1&2\\{ - 1}&3\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}3&{ - 14}\\1&2\end{array}} \right]\end{array}\end{array}\)

\(\begin{array}{l}A \times I = \\\\\left[ {\begin{array}{*{20}{c}}a&b\\c&d\end{array}} \right] \times \left[ {\begin{array}{*{20}{c}}1&0\\0&1\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}a&b\\c&d\end{array}} \right]\\\\I \times A = \\\\\left[ {\begin{array}{*{20}{c}}1&0\\0&1\end{array}} \right] \times \left[ {\begin{array}{*{20}{c}}a&b\\c&d\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}a&b\\c&d\end{array}} \right]\\\\ \Rightarrow \quad A \times I = I \times A = A\end{array}\)

\(\begin{array}{l}A \times (B + C) = \\\\\left[ {\begin{array}{*{20}{c}}1&2\\{ - 1}&0\\2&{ - 1}\end{array}} \right] \times \left( {\left[ {\begin{array}{*{20}{c}}1&{ - 1}\\2&3\end{array}} \right] + \left[ {\begin{array}{*{20}{c}}2&3\\{ - 1}&1\end{array}} \right]} \right) = \\\\ = \left[ {\begin{array}{*{20}{c}}1&2\\{ - 1}&0\\2&{ - 1}\end{array}} \right] \times \left[ {\begin{array}{*{20}{c}}3&2\\1&4\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}5&{10}\\{ - 3}&{ - 2}\\5&0\end{array}} \right]\\\\A \times B + A \times C = \\\\\left[ {\begin{array}{*{20}{c}}1&2\\{ - 1}&0\\2&{ - 1}\end{array}} \right] \times \left[ {\begin{array}{*{20}{c}}1&{ - 1}\\2&3\end{array}} \right] + \left[ {\begin{array}{*{20}{c}}1&2\\{ - 1}&0\\2&{ - 1}\end{array}} \right] \times \left[ {\begin{array}{*{20}{c}}2&3\\{ - 1}&1\end{array}} \right] = \\\\\left[ {\begin{array}{*{20}{c}}5&5\\{ - 1}&1\\0&{ - 5}\end{array}} \right] + \left[ {\begin{array}{*{20}{c}}0&5\\{ - 2}&{ - 3}\\5&5\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}5&{10}\\{ - 3}&{ - 2}\\5&0\end{array}} \right]\\\\ \Rightarrow A \times (B + C) = A \times B + A \times C\end{array}\)

\(\begin{array}{l}A \times (B \times C) = \\\\\left[ {\begin{array}{*{20}{c}}1&2\\{ - 1}&0\\2&{ - 1}\end{array}} \right] \times \left( {\left[ {\begin{array}{*{20}{c}}1&{ - 1}\\2&3\end{array}} \right] \times \left[ {\begin{array}{*{20}{c}}2&3\\{ - 1}&1\end{array}} \right]} \right) = \\\\\left[ {\begin{array}{*{20}{c}}1&2\\{ - 1}&0\\2&{ - 1}\end{array}} \right] \times \left[ {\begin{array}{*{20}{c}}3&2\\1&9\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}5&{20}\\{ - 3}&{ - 2}\\5&{ - 5}\end{array}} \right]\\\\(A \times B) \times C = \\\\\left( {\left[ {\begin{array}{*{20}{c}}1&2\\{ - 1}&0\\2&{ - 1}\end{array}} \right] \times \left[ {\begin{array}{*{20}{c}}1&{ - 1}\\2&3\end{array}} \right]} \right) \times \left[ {\begin{array}{*{20}{c}}2&3\\{ - 1}&1\end{array}} \right] = \\\\ = \left[ {\begin{array}{*{20}{c}}5&5\\{ - 1}&1\\0&{ - 5}\end{array}} \right] \times \left[ {\begin{array}{*{20}{c}}2&3\\{ - 1}&1\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}5&{20}\\{ - 3}&{ - 2}\\5&{ - 5}\end{array}} \right]\\\\ \Rightarrow A \times (B \times C) = (A \times B) \times C\end{array}\)