درسنامه آموزشی فصل اول هندسه (3) کلاس دوازدهم ریاضی

1- اگر $A = \left[ {1\,\,\,\,\,2\,\,\,\,\, - 3} \right]$ و $B = \left[ \begin{gathered}
   - 2 \hfill \\
   - 1 \hfill \\
  \,\,3 \hfill \\ 
\end{gathered}  \right]$ در این صورت $\left| {AB} \right|$ و $\left| {BA} \right|$ را به‌دست آورید.

$A \times B=\left[ {1\,\,\,\,2\,\,\,\,\, - 3} \right] \times \left[ \begin{gathered}
   - 2 \hfill \\
   - 1 \hfill \\
  \,\,3 \hfill \\ 
\end{gathered}  \right] = \left[ { - 13} \right] =  - 13 \Rightarrow \left| {A \times B} \right| =  - 13$

$B \times A = \left[ \begin{gathered}
   - 2 \hfill \\
   - 1 \hfill \\
  \,3 \hfill \\ 
\end{gathered}  \right] \times \left[ {1\,\,\,\,2\,\,\,\, - 3} \right] = \left[ \begin{gathered}
   - 2\,\,\,\,\, - 4\,\,\,\,\, + 6 \hfill \\
   - 1\,\,\,\,\, - 2\,\,\,\,\, + 3 \hfill \\
  \,3\,\,\,\,\,\,\,\,\,6\,\,\,\,\,\,\, - 9 \hfill \\ 
\end{gathered}  \right] \Rightarrow \left| {B \times A} \right| = \left| \begin{gathered}
   - 2\,\,\,\,\, - 4\,\,\,\,\, + 6 \hfill \\
   - 1\,\,\,\,\, - 2\,\,\,\,\, + 3 \hfill \\
  \,\,3\,\,\,\,\,\,\,\,\,6\,\,\,\,\,\, - 9 \hfill \\ 
\end{gathered}  \right|\begin{array}{*{20}{c}}
  { - 2} \\ 
  { - 1} \\ 
  3 
\end{array}\,\,\begin{array}{*{20}{c}}
  { - 4} \\ 
  { - 2} \\ 
  6 
\end{array}$

$ \Rightarrow \left| {B \times A} \right| = \left( {( - 2)( - 2)( - 9) + ( - 4) \times 3 \times 3 + 6 \times ( - 1) \times 6} \right) - $

$\left( {6 \times ( - 2) \times 3 - 2 \times 3 \times 6 + ( - 4)( - 1)( - 9)} \right) =  - 108 + 108 = 0$

2- اگر $A = \left[ \begin{gathered}
   - 2\,\,\,\,\,\,\,\,0\,\,\,\,\,\,\,\,\,\,0 \hfill \\
  \,\,0\,\,\,\,\,\, - 3\,\,\,\,\,\,\,\,\,\,0 \hfill \\
  \,\,1\,\,\,\,\,\,\,\,\,\,0\,\,\,\,\,\,\, - 5 \hfill \\ 
\end{gathered}  \right]$در این صورت $\left| {A'} \right|$ را به‌دست آورید.

${A^2} = \left[ \begin{gathered}
   - 2\,\,\,\,\,0\,\,\,\,\,0 \hfill \\
  0\,\,\,\,\, - 3\,\,\,\,\,0 \hfill \\
  1\,\,\,\,\,0\,\,\,\,\, - 5 \hfill \\ 
\end{gathered}  \right]\left[ \begin{gathered}
   - 2\,\,\,\,\,0\,\,\,\,\,0 \hfill \\
  0\,\,\,\,\, - 3\,\,\,\,\,0 \hfill \\
  1\,\,\,\,\,0\,\,\,\,\, - 5 \hfill \\ 
\end{gathered}  \right] = \left[ \begin{gathered}
  \,\,\,4\,\,\,\,\,0\,\,\,\,\,0 \hfill \\
  \,\,\,0\,\,\,\,\,9\,\,\,\,\,0 \hfill \\
   - 7\,\,\,\,\,0\,\,\,\,\,25 \hfill \\ 
\end{gathered}  \right]$

$ \Rightarrow \left| {{A^2}} \right| = 4 \times 9 \times 25 = 900$

3- اگر $A = \left[ \begin{gathered}
  5\left| A \right|\,\,\,\,\,\,\,\,\left| A \right| \hfill \\
  5\,\,\,\,\,\,\,\,\,\,\,4{\left| A \right|^2} \hfill \\ 
\end{gathered}  \right]$ در این صورت حاصل $\left( {{{\left| A \right|}^3} - 2} \right)$ را بیابید.

$A = \left[ \begin{gathered}
  5\left| A \right|\,\,\,\,\,\,\,\left| A \right| \hfill \\
  \,\,5\,\,\,\,\,\,\,\,4{\left| A \right|^2} \hfill \\ 
\end{gathered}  \right] \Rightarrow \left| A \right| = \left( {5\left| A \right|} \right)\left( {4{{\left| A \right|}^3}} \right) - 5\left| A \right|$

$ \Rightarrow 100{\left| A \right|^3} - 6\left| A \right| = 0$

$ \Rightarrow \left| A \right|\left( {25{{\left| A \right|}^2} - 3} \right) = 0 \Rightarrow \left\{ \begin{gathered}
  \left| A \right| = 0 \Rightarrow {\left| A \right|^3} - 2 =  - 2\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \hfill \\
  {\left| A \right|^2} = \frac{3}{{25}} \Rightarrow \left\{ \begin{gathered}
  \left| A \right| = \frac{{\sqrt 3 }}{5} \Rightarrow {\left| A \right|^3} - 2 = \frac{{\sqrt 3  - 10}}{5} \hfill \\
  \left| A \right| =  - \frac{{\sqrt 3 }}{5} \Rightarrow {\left| A \right|^3} - 2 = \frac{{ - \sqrt 3  - 10}}{5} \hfill \\ 
\end{gathered}  \right. \hfill \\ 
\end{gathered}  \right.$

4- دترمینان ماتریس $A = \left[ \begin{gathered}
  a\,\,\,\,\,b\,\,\,\,\,c \hfill \\
  a\,\,\,\,\,b\,\,\,\,\,c \hfill \\
  d\,\,\,\,\,e\,\,\,\,\,f \hfill \\ 
\end{gathered}  \right]$ را برحسب سطر سوم بیابید. چه نتیجه‌ای می‌گیرید؟

$A = \left[ \begin{gathered}
  a\,\,\,\,\,b\,\,\,\,\,c \hfill \\
  a\,\,\,\,\,b\,\,\,\,\,c \hfill \\
  d\,\,\,\,\,e\,\,\,\,\,f \hfill \\ 
\end{gathered}  \right] \Rightarrow \left| A \right| = {( - 1)^{3 + 1}}d(bc - bc) + {( - 1)^{3 + 2}}e(ac - ac) + {( - 1)^{3 + 3}}f(ab - ab) = 0$

نتیجه: اگر درایه‌های دو سطر (یا دو ستون) یک ماتریس مربعی، نظیر به نظیر مساوی باشند دترمینان آن ماتریس، صفر است.

5- ماتریسی $3 \times 3$ چون $A$ بیابید که $\left| A \right| = 3$.

$A = \left[ \begin{gathered}
  3\,\,\,\,\,\,0\,\,\,\,\,\,\,0 \hfill \\
  0\,\,\,\,\, - 1\,\,\,\,\,0 \hfill \\
  0\,\,\,\,\,0\,\,\,\,\, - 1 \hfill \\ 
\end{gathered}  \right] \Rightarrow \left| A \right| = (3)( - 1)( - 1) = 3$

6- اگر $A = \left[ \begin{gathered}
  4\,\,\,\,\,3 \hfill \\
  2\,\,\,\,\,5 \hfill \\ 
\end{gathered}  \right]$ و $B = \left[ \begin{gathered}
   - 2\,\,\,\,\, - 3 \hfill \\
  5\,\,\,\,\,\,\,\,\, - 1 \hfill \\ 
\end{gathered}  \right]$ حاصل عبارت $(2{A^{ - 1}} - 3{B^{ - 1}})$ را بیابید.

$A = \left[ \begin{gathered}
  4\,\,\,\,\,\,3 \hfill \\
  2\,\,\,\,\,5 \hfill \\ 
\end{gathered}  \right] \Rightarrow \left| A \right| = 20 - 6 = 14 \Rightarrow {A^{ - 1}} = \frac{1}{{14}}\left[ \begin{gathered}
  5\,\,\,\,\, - 3 \hfill \\
   - 2\,\,\,\,\,4 \hfill \\ 
\end{gathered}  \right] = \left[ \begin{gathered}
  \frac{5}{{14}}\,\,\,\,\,\frac{{ - 3}}{{14}} \hfill \\
  \frac{{ - 2}}{{14}}\,\,\,\,\,\frac{4}{{14}} \hfill \\ 
\end{gathered}  \right]$

$B = \left[ \begin{gathered}
   - 2\,\,\,\,\, - 3 \hfill \\
  \,5\,\,\,\,\, - 1 \hfill \\ 
\end{gathered}  \right] \Rightarrow \left| B \right| = 2 + 15 = 17 \Rightarrow {B^{ - 1}} = \frac{1}{{17}}\left[ \begin{gathered}
   - 1\,\,\,\,\,\,\,\,3 \hfill \\
   - 5\,\,\,\,\, - 2 \hfill \\ 
\end{gathered}  \right] = \left[ \begin{gathered}
  \frac{{ - 1}}{{17}}\,\,\,\,\,\frac{3}{{17}} \hfill \\
  \frac{{ - 5}}{{17}}\,\,\,\,\,\frac{{ - 2}}{{17}} \hfill \\ 
\end{gathered}  \right]$

$2{A^{ - 1}} - 3{B^{ - 1}} = \left[ \begin{gathered}
  \frac{5}{7}\,\,\,\,\,\frac{{ - 3}}{7} \hfill \\
  \frac{{ - 2}}{7}\,\,\,\,\,\frac{4}{7} \hfill \\ 
\end{gathered}  \right] + \left[ \begin{gathered}
  \frac{{ + 3}}{{17}}\,\,\,\,\,\frac{{ - 9}}{{17}} \hfill \\
  \frac{{15}}{{17}}\,\,\,\,\,\,\frac{6}{{17}} \hfill \\ 
\end{gathered}  \right] = \left[ \begin{gathered}
  \frac{{106}}{{119}}\,\,\,\,\,\, - \frac{{114}}{{119}} \hfill \\
  \frac{{71}}{{119}}\,\,\,\,\,\,\,\,\,\frac{{110}}{{119}} \hfill \\ 
\end{gathered}  \right]$

7- اگر $A = \left[ \begin{gathered}
  5\,\,\,\,\,\,2 \hfill \\
  3\,\,\,\,\,\,2 \hfill \\ 
\end{gathered}  \right]$ ابتدا ماتریس ${A^{ - 1}}$ را به‌دست آورده و $\left| A \right|$ را با $\left| {{A^{ - 1}}} \right|$ مقایسه کنید.

$A = \left[ \begin{gathered}
  5\,\,\,\,\,2 \hfill \\
  3\,\,\,\,\,2 \hfill \\ 
\end{gathered}  \right] \Rightarrow \left| A \right| = 10 - 6 = 4 \Rightarrow {A^{ - 1}} = \frac{1}{4}\left[ \begin{gathered}
  2\,\,\,\,\, - 2 \hfill \\
   - 3\,\,\,\,\,\,5 \hfill \\ 
\end{gathered}  \right]$

$ = \left[ \begin{gathered}
  \,\,\frac{1}{2}\,\,\,\,\,\, - \frac{1}{2} \hfill \\
   - \frac{3}{4}\,\,\,\,\,\,\,\,\,\frac{5}{4} \hfill \\ 
\end{gathered}  \right] \Rightarrow \left| {{A^{ - 1}}} \right| = \frac{5}{8} - \frac{3}{8} = \frac{1}{4} \Rightarrow \left| {{A^{ - 1}}} \right| = \frac{1}{{\left| A \right|}}$

8- الف) ماتریس‌های $A = \left[ \begin{gathered}
  a\,\,\,\,\,\,b\,\,\,\,\,\,\,c \hfill \\
  d\,\,\,\,\,\,e\,\,\,\,\,\,f \hfill \\
  g\,\,\,\,\,\,h\,\,\,\,\,\,i \hfill \\ 
\end{gathered}  \right]$ و $(k \in \mathbb{R})B = \left[ \begin{gathered}
  ka\,\,\,\,\,\,kb\,\,\,\,\,\,kc \hfill \\
  \,d\,\,\,\,\,\,\,\,\,e\,\,\,\,\,\,\,\,f \hfill \\
  g\,\,\,\,\,\,\,\,h\,\,\,\,\,\,\,\,\,i \hfill \\ 
\end{gathered}  \right]$ را در نظر بگیرید و $\left| A \right|$ و $\left| B \right|$ را از دستور ساروس محاسبه کرده و با هم مقایسه کنید. چه نتیجه‌ای می‌گیرید؟

$A = \left[ \begin{gathered}
  a\,\,\,\,\,b\,\,\,\,\,c \hfill \\
  d\,\,\,\,\,e\,\,\,\,\,f \hfill \\
  g\,\,\,\,\,h\,\,\,\,\,i \hfill \\ 
\end{gathered}  \right]\begin{array}{*{20}{c}}
  a \\ 
  d \\ 
  g 
\end{array}\,\,\begin{array}{*{20}{c}}
  b \\ 
  e \\ 
  h 
\end{array} \Rightarrow \left| A \right| = (aei + bfg + cdh) - (bdi + afh + ceg)$

$B = \left[ \begin{gathered}
  ka\,\,\,\,\,kb\,\,\,\,\,kc \hfill \\
  \,d\,\,\,\,\,\,\,\,e\,\,\,\,\,\,\,f \hfill \\
  g\,\,\,\,\,\,\,h\,\,\,\,\,\,\,\,i \hfill \\ 
\end{gathered}  \right]\begin{array}{*{20}{c}}
  {ka} \\ 
  d \\ 
  g 
\end{array}\,\,\begin{array}{*{20}{c}}
  {kb} \\ 
  e \\ 
  h 
\end{array} \Rightarrow \left| B \right| = (kaei + kbfg + kcdh) - (kbdi + kafh + kceg) \Rightarrow \left| B \right| = k\left| A \right|$

ب) قسمت (الف) را برای دو ماتریس $A = \left[ \begin{gathered}
  a\,\,\,\,b \hfill \\
  c\,\,\,\,d \hfill \\ 
\end{gathered}  \right]$ و $(k \in \mathbb{R})B = \left[ \begin{gathered}
  ka\,\,\,\,kb \hfill \\
  \,c\,\,\,\,\,\,\,d \hfill \\ 
\end{gathered}  \right]$ بررسی کنید.

$A = \left[ \begin{gathered}
  a\,\,\,\,\,b \hfill \\
  c\,\,\,\,\,d \hfill \\ 
\end{gathered}  \right]\,\, \Rightarrow \left| A \right| = ad - bc\,,B = \left[ \begin{gathered}
  ka\,\,\,\,kb \hfill \\
  \,\,c\,\,\,\,\,\,\,d \hfill \\ 
\end{gathered}  \right] \Rightarrow \left| B \right| = kad - kbc \Rightarrow \left| B \right| = k\left| A \right|$

9- برای ماتریس $2 \times 2$ مانند $A$ دو مقدار $\left| A \right|$ و $(k \in \mathbb{R})\left| {KA} \right|$ را با هم مقایسه کنید. چه نتیجه‌ای می‌گیرید؟

$A = \left[ \begin{gathered}
  a\,\,\,\,\,b \hfill \\
  c\,\,\,\,\,d \hfill \\ 
\end{gathered}  \right] \Rightarrow \left| A \right| = ad - bc\,,\,KA = \left[ \begin{gathered}
  ka\,\,\,\,\,kb \hfill \\
  kc\,\,\,\,\,kd \hfill \\ 
\end{gathered}  \right] \Rightarrow $

$\left| {KA} \right| = (ka)(kd) - (kb)(kc) = {k^2}ad - {k^2}bc \Rightarrow \left| {KA} \right| = {k^2}\left| A \right|$

10- اگر $A$ ماتریسی $3 \times 3$ باشد و $\left| A \right| = 5$ در این صورت حاصل $\left| {\left| A \right|A} \right|$ را بیابید.

$\left| {{A_{3 \times 3}}} \right| = 5 \Rightarrow \left| {\left| A \right|A} \right| = \left| {5A} \right| = {5^3}\left| A \right| = {5^3} \times 5 = {5^4} = 625$

11- دستگاه معادلات خطی تشکیل دهید که $A = \left[ \begin{gathered}
  3\,\,\,\,\, - 5 \hfill \\
  4\,\,\,\,\,\,\,\,2 \hfill \\ 
\end{gathered}  \right]$ ماتریس ضرایب دستگاه بوده و $B = \left[ \begin{gathered}
  \,1 \hfill \\
  10 \hfill \\ 
\end{gathered}  \right]$ ماتریس معلومات آن باشد و سپس جواب دستگاه را با استفاده از ${A^{ - 1}}$ بیابید.

$\left\{ \begin{gathered}
  3x - 5y = 1\,\,\,\, \hfill \\
  4x + 2y = 10 \hfill \\ 
\end{gathered}  \right.$

$\left| A \right| = 6 + 20 = 26 \ne 0 \Rightarrow {A^{ - 1}} = \frac{1}{{26}}\left[ \begin{gathered}
  2\,\,\,\,\,\,\,\,5 \hfill \\
   - 4\,\,\,\,3 \hfill \\ 
\end{gathered}  \right]$

$X = {A^{ - 1}} \times B \Rightarrow \left[ \begin{gathered}
  x \hfill \\
  y \hfill \\ 
\end{gathered}  \right] = \frac{1}{{26}}\left[ \begin{gathered}
  2\,\,\,\,\,\,5 \hfill \\
   - 4\,\,\,3 \hfill \\ 
\end{gathered}  \right] \times \left[ \begin{gathered}
  \,\,1 \hfill \\
  10 \hfill \\ 
\end{gathered}  \right]$

$ = \frac{1}{{26}}\left[ \begin{gathered}
  52 \hfill \\
  26 \hfill \\ 
\end{gathered}  \right] = \left[ \begin{gathered}
  \frac{{52}}{{26}} \hfill \\
  \frac{{26}}{{26}} \hfill \\ 
\end{gathered}  \right] = \left[ \begin{gathered}
  2 \hfill \\
  1 \hfill \\ 
\end{gathered}  \right] \Rightarrow \left\{ \begin{gathered}
  x = 2 \hfill \\
  y = 1 \hfill \\ 
\end{gathered}  \right.$

12- به ازای چه مقادیری از $k$ دستگاه $\left\{ \begin{gathered}
  kx + 3y = 4 \hfill \\
  x - 2y = 3\,\,\, \hfill \\ 
\end{gathered}  \right.$ یک دسته جواب منحصر به فرد دارد.

$ \Rightarrow \frac{k}{1} \ne  - \frac{3}{2} \Rightarrow k \ne  - \frac{3}{2}$

13- روی وجود و عدم وجود و تعداد جواب‌های هر یک از دستگاه‌های زیر بحث کنید و در صورت وجود، جواب را با استفاده از ${A^{ - 1}}$ بیابید.

$\left\{ \begin{gathered}
  3x - 5y =  - 1 \hfill \\
  2x + y = 8\,\,\,\,\, \hfill \\ 
\end{gathered}  \right.$ (الف

دستگاه فقط یک دسته جواب منحصر به فرد دارد. $ \Rightarrow \frac{3}{2} \ne  - \frac{5}{1} \Rightarrow $

$\left\{ \begin{gathered}
  x + 3y = 5\,\,\,\,\,\, \hfill \\
   - 2x - 6y = 1\,\,\,\,\, \hfill \\ 
\end{gathered}  \right.$ (ب

$ \Rightarrow  - \frac{1}{2} =  - \frac{3}{6} \ne \frac{5}{1}$

$\left\{ \begin{gathered}
   - 2x + 3y = 2 \hfill \\
  4x - 6y =  - 4\,\,\,\,\, \hfill \\ 
\end{gathered}  \right.$ (پ

دستگاه بی‌شمار جواب دارد.   $ \Rightarrow  - \frac{2}{4} =  - \frac{3}{6} =  - \frac{2}{4} \Rightarrow $