گام به گام تمرین صفحه 20 درس 1 هندسه دوازدهم (ماتریس و کاربردها)

1)

\(\begin{array}{l}A = {\left[ {{a_{ij}}} \right]_{3 \times 4}} = \left\{ \begin{array}{l}7\quad \quad \quad i = j\\i + j\quad \;\;\,{\kern 1pt} i > j\\{i^2}\quad \quad \;\;\,{\kern 1pt} i < j\end{array} \right.\quad \\\\ \Rightarrow \quad A = \left[ {\begin{array}{*{20}{c}}7&1&1&1\\3&7&4&4\\4&5&7&9\end{array}} \right]\end{array}\)

2)

\(\begin{array}{l}\underline {\left\{ \begin{array}{l}2x - y = 3\\2x + y = 5\end{array} \right.} \\\\4x = 8 \Rightarrow x = 2 \Rightarrow y = 1 \Rightarrow z =  - 2\\ \Rightarrow x + y + z = 1\end{array}\)

3)

\(\left\{ \begin{array}{l}A = \left[ {\begin{array}{*{20}{c}}2&3&0\\4&6&0\\0&0&0\end{array}} \right] \ne \overline O \\B = \left[ {\begin{array}{*{20}{c}}0&3&{ - 6}\\0&{ - 2}&4\\2&4&5\end{array}} \right] \ne \overline O \end{array} \right.\quad \\\\ \Rightarrow \quad A \times B = \left[ {\begin{array}{*{20}{c}}0&0&0\\0&0&0\\0&0&0\end{array}} \right] = \overline O\)

4)

مثال نقض اول:

\(\begin{array}{l}A = \left[ {\begin{array}{*{20}{c}}1&2\\2&4\end{array}} \right]\;,\;B = \left[ {\begin{array}{*{20}{c}}{ - 1}&2\\2&1\end{array}} \right]\;,\;C = \left[ {\begin{array}{*{20}{c}}{ - 3}&8\\3&{ - 2}\end{array}} \right]\quad \\ \Rightarrow \quad AB = AC = \left[ {\begin{array}{*{20}{c}}3&4\\6&8\end{array}} \right]\quad ;\quad B \ne C\end{array}\)

مثال نقض دوم :

\(\begin{array}{l}A = \left[ {\begin{array}{*{20}{c}}1&2&1\\1&0&{ - 1}\\1&{ - 2}&{ - 3}\end{array}} \right]\;,\;B = \left[ {\begin{array}{*{20}{c}}1&{ - 1}&1\\0&1&0\\1&1&0\end{array}} \right]\;,\;C = \left[ {\begin{array}{*{20}{c}}2&1&2\\{ - 1}&{ - 1}&{ - 1}\\2&3&1\end{array}} \right]\\\quad  \Rightarrow \quad AB = AC = \left[ {\begin{array}{*{20}{c}}2&2&1\\0&{ - 2}&1\\{ - 2}&{ - 6}&1\end{array}} \right]\quad ;\quad B \ne C\end{array}\)

5)

\(\begin{array}{l}\left\{ \begin{array}{l}A = \left[ {\begin{array}{*{20}{c}}1&0\\0&{ - 1}\end{array}} \right] \Rightarrow {A^2} = \left[ {\begin{array}{*{20}{c}}1&0\\0&{ - 1}\end{array}} \right] \times \left[ {\begin{array}{*{20}{c}}1&0\\0&{ - 1}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}1&0\\0&1\end{array}} \right] = I\\{A^3} = A.{A^2} = \left[ {\begin{array}{*{20}{c}}1&0\\0&{ - 1}\end{array}} \right] \times \left[ {\begin{array}{*{20}{c}}1&0\\0&1\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}1&0\\0&{ - 1}\end{array}} \right]\end{array} \right.\\\\ \Rightarrow \left\{ \begin{array}{l}A = {A^3} = {A^5} =  \cdots  = \left[ {\begin{array}{*{20}{c}}1&0\\0&{ - 1}\end{array}} \right]\\{A^2} = {A^4} = {A^6} =  \cdots  = \left[ {\begin{array}{*{20}{c}}1&0\\0&1\end{array}} \right] = I\end{array} \right.\end{array}\)

6)

\(\begin{array}{l}A \times B = \left[ {\begin{array}{*{20}{c}}4&a\\b&{ - 1}\end{array}} \right] \times \left[ {\begin{array}{*{20}{c}}1&{ - 2}\\3&2\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{4 + 3a}&{ - 8 + 2a}\\{b - 3}&{ - 2b + 2}\end{array}} \right]\\\\ \Rightarrow \left\{ \begin{array}{l} - 8 + 2a = 0\\b - 3 = 0\end{array} \right. \Rightarrow \left\{ \begin{array}{l}a = 4\\b = 3\end{array} \right.\end{array}\)

7)

\(\begin{array}{l}A = \left[ {\begin{array}{*{20}{c}}0&1\\1&3\\2&1\end{array}} \right]\quad ,\quad B = \left[ {\begin{array}{*{20}{c}}2&1&0\\2&5&1\end{array}} \right]\quad \\\\ \Rightarrow \;A \times B = \left[ {\begin{array}{*{20}{c}}2&5&1\\8&{16}&3\\6&7&1\end{array}} \right]\;\;,\;\;B \times A = \left[ {\begin{array}{*{20}{c}}1&5\\7&{18}\end{array}} \right]\end{array}\)

8)

\(\begin{array}{l}A = \left[ {\begin{array}{*{20}{c}}{{r_1}}&0&0\\0&{{r_2}}&0\\0&0&{{r_3}}\end{array}} \right]\quad ,\quad B = \left[ {\begin{array}{*{20}{c}}{{a_{11}}}&{{a_{12}}}&{{a_{13}}}\\{{a_{21}}}&{{a_{22}}}&{{a_{23}}}\\{{a_{31}}}&{{a_{32}}}&{{a_{33}}}\end{array}} \right]\quad \\\\ \Rightarrow \quad A \times B = \left[ {\begin{array}{*{20}{c}}{{r_1}{a_{11}}}&{{r_1}{a_{12}}}&{{r_1}{a_{13}}}\\{{r_2}{a_{21}}}&{{r_2}{a_{22}}}&{{r_2}{a_{23}}}\\{{r_3}{a_{31}}}&{{r_3}{a_{32}}}&{{r_3}{a_{33}}}\end{array}} \right]\end{array}\)

نتیجه : اگر A ماتریسی قطری و B یک ماتریس مربعی هم مرتیه با A باشد. برای محاسبه \(A \times B\) کافی است درایه های قطر اصلی A را در  درایه های سطرهای نظیر آن ها در  ماتریس B ضرب کنیم.

9)

اگر \(B = \left[ {\begin{array}{*{20}{c}}{{a_{11}}}&{{a_{12}}}&{{a_{13}}}\\{{a_{21}}}&{{a_{22}}}&{{a_{23}}}\\{{a_{31}}}&{{a_{32}}}&{{a_{33}}}\end{array}} \right]\) و \(A = \left[ {\begin{array}{*{20}{c}}r&0&0\\0&r&0\\0&0&r\end{array}} \right]\)، آنگاه :

\(\begin{array}{l}A \times B = \left[ {\begin{array}{*{20}{c}}{r{a_{11}}}&{r{a_{12}}}&{r{a_{13}}}\\{r{a_{21}}}&{r{a_{22}}}&{r{a_{23}}}\\{r{a_{31}}}&{r{a_{32}}}&{r{a_{33}}}\end{array}} \right] = rB\quad ,\quad \\\\B \times A = \left[ {\begin{array}{*{20}{c}}{r{a_{11}}}&{r{a_{12}}}&{r{a_{13}}}\\{r{a_{21}}}&{r{a_{22}}}&{r{a_{23}}}\\{r{a_{31}}}&{r{a_{32}}}&{r{a_{33}}}\end{array}} \right] = rB\\\\ \Rightarrow A \times B = B \times A = rB\end{array}\)

10)

الف)

\(\begin{array}{l}{(A + B)^2} = (A + B)(A + B) = (A + B)A + (A + B)B\\ = {A^2} + BA + AB + {B^2}\; = {A^2} + AB + AB + {B^2}\\ = {A^2} + 2AB + {B^2}\end{array}\)

ب)

\(\begin{array}{l}(A + B)(A - B) = (A + B)A\\ - (A + B)B = {A^2} + BA - AB - {B^2}\\ = {A^2} + AB - AB - {B^2} = {A^2} - {B^2}\end{array}\)

11)

\(\begin{array}{l}A = \left[ {\begin{array}{*{20}{c}}{ - 2}&0&0\\0&3&0\\0&0&4\end{array}} \right] \Rightarrow \\\\{A^2} = \left[ {\begin{array}{*{20}{c}}{ - 2}&0&0\\0&3&0\\0&0&4\end{array}} \right] \times \left[ {\begin{array}{*{20}{c}}{ - 2}&0&0\\0&3&0\\0&0&4\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}4&0&0\\0&9&0\\0&0&{16}\end{array}} \right]\\\\ \Rightarrow {A^3} = \left[ {\begin{array}{*{20}{c}}4&0&0\\0&9&0\\0&0&{16}\end{array}} \right] \times \left[ {\begin{array}{*{20}{c}}{ - 2}&0&0\\0&3&0\\0&0&4\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{ - 8}&0&0\\0&{27}&0\\0&0&{64}\end{array}} \right]\\\\A = {\left[ {\begin{array}{*{20}{c}}{{a_{11}}}&0& \cdots &0\\0&{{a_{22}}}& \cdots &0\\ \vdots & \vdots & \ddots & \vdots \\0&0& \cdots &{{a_{nn}}}\end{array}} \right]_{n \times n}}\quad ,\;\;k \in \mathbb{N}\\\\ \Rightarrow {A^k} = {\left[ {\begin{array}{*{20}{c}}{{a_{11}}^k}&0& \cdots &0\\0&{{a_{22}}^k}& \cdots &0\\ \vdots & \vdots & \ddots & \vdots \\0&0& \cdots &{{a_{nn}}^k}\end{array}} \right]_{n \times n}}\end{array}\)