جواب کار در کلاس صفحه 29 درس 1 هندسه دوازدهم (ماتریس و کاربردها)

\(\begin{array}{l}A \times B = \\\\\left[ {\begin{array}{*{20}{c}}2&{ - 1}\\3&4\end{array}} \right] \times \left[ {\begin{array}{*{20}{c}}3&1\\4&2\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}2&0\\{25}&{11}\end{array}} \right]\\\\ \Rightarrow \left| {AB} \right| = 2 \times 11 - 0 \times 25 = 22\\\\\left| A \right| = 2 \times 4 - 3\left( { - 1} \right) = 11\\\\\left| B \right| = 3 \times 2 - 4 \times 1 = 2\\\\ \Rightarrow \left| A \right|\left| B \right| = 11 \times 2 = 22\\\\ \Rightarrow \left| {AB} \right| = \left| A \right|\left| B \right|\end{array}\)

برای این دو ماتریس، تساوی مذکور برقرار است.

\(\begin{array}{l}A = \left[ {\begin{array}{*{20}{c}}2&0&0\\1&{ - 1}&0\\2&0&3\end{array}} \right]\\\\\left| A \right| = {\left( { - 1} \right)^{1 + 1}} \times 2 \times \left| {\begin{array}{*{20}{c}}{ - 1}&0\\0&3\end{array}} \right| + 0 + 0\\\\ \Rightarrow \left| A \right| = - 6\\\\ \Rightarrow {\left| A \right|^2} = {\left( { - 6} \right)^2} = 36\\\\{A^2} = \left[ {\begin{array}{*{20}{c}}2&0&0\\1&{ - 1}&0\\2&0&3\end{array}} \right] \times \left[ {\begin{array}{*{20}{c}}2&0&0\\1&{ - 1}&0\\2&0&3\end{array}} \right] = \\\\\left[ {\begin{array}{*{20}{c}}4&0&0\\1&1&0\\{10}&0&9\end{array}} \right]\\\\\left| {{A^2}} \right| = {\left( { - 1} \right)^{1 + 1}} \times 4 \times \left| {\begin{array}{*{20}{c}}1&0\\0&9\end{array}} \right| + 0 + 0\\\\ \Rightarrow \left| {{A^2}} \right| = 36\\\\ \Rightarrow \left| {{A^2}} \right| = {\left| A \right|^2}\end{array}\)

\(\begin{array}{l}A = \left[ {\begin{array}{*{20}{c}}a&0&0\\0&b&0\\0&0&c\end{array}} \right]\\\\\left| A \right| = {\left( { - 1} \right)^{1 + 1}} \times a \times \left| {\begin{array}{*{20}{c}}b&0\\0&c\end{array}} \right| + 0 + 0\\\\ \Rightarrow \left| A \right| = abc\end{array}\)

\(\begin{array}{*{20}{l}}{A = 4\left[ {\begin{array}{*{20}{c}}{\frac{1}{2}}&0&0\\0&{\frac{5}{4}}&0\\0&0&{\frac{5}{2}}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}2&0&0\\0&5&0\\0&0&{10}\end{array}} \right]}\\{}\\{ \Rightarrow \left\{ {\begin{array}{*{20}{l}}\begin{array}{l}(1):\left| A \right| = 2 \times 5 \times 10 = 100\\\end{array}\\{(2):\left| A \right| = {4^3} \times \frac{1}{2} \times \frac{5}{4} \times \frac{5}{2} = 100}\end{array}} \right.}\end{array}\)