بين أن الخصائص الآتية صحيحة للمصفوفات من النوع 2×2.

الخاصية التجميعية للضرب في عدد.

$c\left(\left[\begin{array}{l}{a}_{11}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{a}_{12}\\ a_{21}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{a}_{22}\end{array}\right]\left[\begin{array}{l}{b}_{11}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{b}_{12}\\ b_{21}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{b}_{22}\end{array}\right]\right)=c\left(\underset{\_}{A}\underset{\_}{B}\right)$ بالتعويض.

$c\left[\begin{array}{cc}{a}_{11}{b}_{11}+a_{12}{b}_{21}& a_{11}{b}_{12}+a_{12}{b}_{22}\\ a_{21}{b}_{11}+a_{22}{b}_{21}& a_{21}{b}_{12}+a_{22}{b}_{22}\end{array}\right]=$ تعريف ضرب المصفوفات.

$\left[\begin{array}{l}c\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\left(a_{11}{b}_{11}+a_{12}{b}_{21}\right)\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}c\left(a_{11}{b}_{12}+a_{12}{b}_{22}\right)\\ c\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\left(a_{21}{b}_{11}+a_{22}{b}_{21}\right)\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}c\left(a_{21}{b}_{12}+a_{22}{b}_{22}\right)\end{array}\right]=$ تعريف الضرب في عدد ثابت.

$\left[\begin{array}{cc}\begin{array}{ll}c{a}_{11}{b}_{11}+c{a}_{12}{b}_{21}& c{a}_{11}{b}_{12}+c{a}_{12}{b}_{22}\\ c{a}_{21}{b}_{11}+c{a}_{22}{b}_{21}& c{a}_{21}{b}_{12}+c{a}_{22}{b}_{22}\end{array}& \end{array}\right]=$ خاصية التوزيع.

$\left[\begin{array}{cc}\begin{array}{ll}c{a}_{11}& c{a}_{12}\\ c{a}_{21}& c{a}_{22}\end{array}& \end{array}\right]\left[\begin{array}{cc}\begin{array}{ll}{b}_{11}& b_{12}\\ b_{21}& b_{22}\end{array}& \end{array}\right]=$ تعريف ضرب المصفوفات.

$c\left[\begin{array}{cc}\begin{array}{ll}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}& \end{array}\right]\left[\begin{array}{cc}\begin{array}{ll}{b}_{11}& b_{12}\\ b_{21}& b_{22}\end{array}& \end{array}\right]=$ تعريف الضرب في عدد ثابت.

$\left(c\underset{\_}{A}\right)\underset{\_}{B}=$ بالتعويض.

وبالمثل $c\left(\underset{\_}{A}\underset{\_}{B}\right)=\underset{\_}{A}\left(c\underset{\_}{B}\right)$

حل أسئلة مسائل مهارات التفكير العليا

43) تبرير: إذا كانت رتبة المصفوفة AB هي 8×5 ورتبة المصفوفة A هي 6×5 فما رتبة المصفوفة B؟

8 $\times$ 6

44) برهان: بين أن الخصائص الآتية صحيحة للمصفوفات من النوع 2×2.

a) خاصية التوزيع للضرب في عدد.

$c\left(\begin{array}{cc}\left[\begin{array}{cc}\begin{array}{ll}a& b\\ d& e\end{array}& \end{array}\right]+& \left[\begin{array}{cc}\begin{array}{ll}w& x\\ y& z\end{array}& \end{array}\right]\end{array}\right)=c(\underset{\_}{A}+\underset{\_}{B})$ بالتعويض.

$c\left[\begin{array}{cc}\begin{array}{ll}a+w& b+x\\ d+y& e+z\end{array}& \end{array}\right]=$ تعريف جمع المصفوفات.

$\left[\begin{array}{cc}\begin{array}{ll}ca+cw& cb+cx\\ cd+cy& ce+cz\end{array}& \end{array}\right]=$ تعريف ضرب المصفوفات في ثابت.

$\left[\begin{array}{cc}\begin{array}{ll}ca& cb\\ cd& ce\end{array}& \end{array}\right]+\left[\begin{array}{cc}\begin{array}{ll}cw& cx\\ cy& cz\end{array}& \end{array}\right]=$ تعريف جمع المصفوفات.

$cA+cB=$ بالتعويض.

b) خاصية التوزيع للضرب على الجمع.

$\left[\begin{array}{cc}\begin{array}{ll}a& b\\ c& d\end{array}& \end{array}\right]\left(\begin{array}{cc}\left[\begin{array}{cc}\begin{array}{ll}e& f\\ g& h\end{array}& \end{array}\right]+& \left[\begin{array}{cc}\begin{array}{cc}j& k\\ m& n\end{array}& \end{array}\right]\end{array}\right)=\underset{\_}{C}(\underset{\_}{A}+\underset{\_}{B})$ بالتعويض.

$\left[\begin{array}{cc}\begin{array}{ll}a& b\\ c& d\end{array}& \end{array}\right]\left[\begin{array}{cc}\begin{array}{cc}e+j& f+k\\ g+m& h+n\end{array}& \end{array}\right]=$ تعريف جمع المصفوفات.

$\left[\begin{array}{cc}\begin{array}{ll}a(e+j)+b(g+m)& a(f+k)+b(h+n)\\ c(e+j)+d(g+m)& c(f+k)+d(h+n)\end{array}& \end{array}\right]=$ تعريف ضرب المصفوفات.

$\left[\begin{array}{cc}\begin{array}{ll}ea+ja+gb+mb& fa+ka+hb+nb\\ ec+jc+gd+md& fc+kc+hd+nd\end{array}& \end{array}\right]=$ خاصية التوزيع.

$\left[\begin{array}{cc}\begin{array}{ll}ea+gb+ja+mb& fa+hb+ka+nb\\ ec+gd+jc+md& fc+hd+kc+nd\end{array}& \end{array}\right]=$ خاصية الإبدال على الجمع.

$\left[\begin{array}{cc}\begin{array}{ll}ea+gb& fa+hb\\ ec+gd& fc+hd\end{array}& \end{array}\right]+\left[\begin{array}{cc}\begin{array}{ll}ja+mb& ka+nb\\ jc+md& kc+nd\end{array}& \end{array}\right]=$ خاصية الجمع على المصفوفات.

$\underset{\_}{C}\underset{\_}{A}+\underset{\_}{C}\underset{\_}{B}=$ تعريف جمع المصفوفات.

وبالمثل $\underset{\_}{A}\underset{\_}{C}+\underset{\_}{B}\underset{\_}{C}=(\underset{\_}{A}+\underset{\_}{B})\underset{\_}{C}$

c) الخاصية التجميعية للضرب.

$\left(\begin{array}{cc}\left[\begin{array}{cc}\begin{array}{ll}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}& \end{array}\right]& \left[\begin{array}{cc}\begin{array}{ll}{b}_{11}& b_{21}\\ b_{21}& b_{22}\end{array}& \end{array}\right]\end{array}\right)\left[\begin{array}{cc}\begin{array}{ll}{c}_{11}& c_{12}\\ c_{21}& c_{22}\end{array}& \end{array}\right]=\left(\underset{\_}{AB}\right)\underset{\_}{C}$ بالتعويض.

$\left[\begin{array}{cc}\begin{array}{ll}{a}_{11}{b}_{11}+a_{12}{b}_{21}& a_{11}{b}_{12}+a_{12}{b}_{22}\\ a_{21}{b}_{11}+a_{22}{b}_{21}& a_{21}{b}_{12}+a_{22}{b}_{22}\end{array}& \end{array}\right]\left[\begin{array}{cc}\begin{array}{ll}{c}_{11}& c_{12}\\ c_{21}& c_{22}\end{array}& \end{array}\right]=$ تعريف ضرب المصفوفات.

$\left[\begin{array}{cc}\begin{array}{ll}(a_{11}{b}_{11}+a_{12}{b}_{21})c_{11}+(a_{11}{b}_{12}+a_{12}{b}_{22})c_{21}& (a_{11}{b}_{11}+a_{12}{b}_{21})c_{12}+(a_{11}{b}_{12}+a_{12}{b}_{22})c_{22}\\ (a_{21}{b}_{11}+a_{22}{b}_{21})c_{11}+(a_{21}{b}_{12}+a_{22}{b}_{22})c_{21}& (a_{21}{b}_{11}+a_{22}{b}_{21})c_{12}+(a_{21}{b}_{12}+a_{22}{b}_{22})c_{22}\end{array}& \end{array}\right]=$ خاصية التوزيع.

$\left[\begin{array}{cc}\begin{array}{ll}{a}_{11}{b}_{11}{c}_{11}+a_{12}{b}_{21}{c}_{11}+a_{11}{b}_{12}{c}_{21}+a_{12}{b}_{22}{c}_{21}& a_{11}{b}_{11}{c}_{12}+a_{12}{b}_{21}{c}_{12}+a_{11}{b}_{12}{c}_{22}+a_{12}{b}_{22}{c}_{22}\\ a_{21}{b}_{11}{c}_{11}+a_{22}{b}_{21}{c}_{11}+a_{21}{b}_{12}{c}_{21}+a_{22}{b}_{22}{c}_{21}& a_{21}{b}_{11}{c}_{12}+a_{22}{b}_{21}{c}_{12}+a_{21}{b}_{12}{c}_{22}+a_{22}{b}_{22}{c}_{22}\end{array}& \end{array}\right]=$ خاصية الإبدال على الجمع.

$\left[\begin{array}{cc}\begin{array}{ll}{a}_{11}{b}_{11}{c}_{11}+a_{11}{b}_{12}{c}_{21}+a_{12}{b}_{21}{c}_{11}+a_{12}{b}_{22}{c}_{21}& a_{11}{b}_{11}{c}_{12}+a_{11}{b}_{12}{c}_{22}+a_{12}{b}_{21}{c}_{12}+a_{12}{b}_{22}{c}_{22}\\ a_{21}{b}_{11}{c}_{11}+a_{21}{b}_{12}{c}_{21}+a_{22}{b}_{21}{c}_{11}+a_{22}{b}_{22}{c}_{21}& a_{21}{b}_{11}{c}_{12}+a_{21}{b}_{12}{c}_{22}+a_{22}{b}_{21}{c}_{12}+a_{22}{b}_{22}{c}_{22}\end{array}& \end{array}\right]=$ تعريف جمع المصفوفات.

$\left[\begin{array}{cc}\begin{array}{ll}{a}_{11}(b_{11}{c}_{11}+b_{12}{c}_{21})+a_{12}(b_{21}{c}_{11}+b_{22}{c}_{21})& a_{11}(b_{11}{c}_{12}+b_{12}{c}_{22})+a_{12}(b_{21}{c}_{12}+b_{22}{c}_{22})\\ a_{21}(b_{11}{c}_{11}+b_{12}{c}_{21})+a_{22}(b_{21}{c}_{11}+b_{22}{c}_{21})& a_{21}(b_{11}{c}_{12}+b_{12}{c}_{22})+a_{22}(b_{21}{c}_{12}+b_{22}{c}_{22})\end{array}& \end{array}\right]=$

$\left[\begin{array}{cc}\begin{array}{ll}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}& \end{array}\right]\left(\begin{array}{cc}\left[\begin{array}{cc}\begin{array}{ll}{b}_{11}{c}_{11}+b_{12}{c}_{21}& b_{11}{c}_{12}+b_{12}{c}_{22}\\ b_{21}{c}_{11}+b_{22}{c}_{21}& b_{21}{c}_{12}+b_{22}{c}_{22}\end{array}& \end{array}\right]& \end{array}\right)=$ تعريف ضرب المصفوفات.

$\left[\begin{array}{cc}\begin{array}{ll}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}& \end{array}\right]\left(\begin{array}{cc}\left[\begin{array}{cc}\begin{array}{ll}{b}_{11}& b_{12}\\ b_{21}& b_{22}\end{array}& \end{array}\right]& \left[\begin{array}{cc}\begin{array}{ll}{c}_{11}& c_{12}\\ c_{21}& c_{22}\end{array}& \end{array}\right]\end{array}\right)=$ تعريف ضرب المصفوفات.

$\underset{\_}{A}\left(\underset{\_}{B}\underset{\_}{C}\right)=$ بالتعويض.

d) الخاصية التجميعية للضرب في عدد.

$c\left(\left[\begin{array}{l}{a}_{11}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{a}_{12}\\ a_{21}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{a}_{22}\end{array}\right]\left[\begin{array}{l}{b}_{11}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{b}_{12}\\ b_{21}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{b}_{22}\end{array}\right]\right)=c\left(\underset{\_}{A}\underset{\_}{B}\right)$ بالتعويض.

$c\left[\begin{array}{cc}{a}_{11}{b}_{11}+a_{12}{b}_{21}& a_{11}{b}_{12}+a_{12}{b}_{22}\\ a_{21}{b}_{11}+a_{22}{b}_{21}& a_{21}{b}_{12}+a_{22}{b}_{22}\end{array}\right]=$ تعريف ضرب المصفوفات.

$\left[\begin{array}{l}c\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\left(a_{11}{b}_{11}+a_{12}{b}_{21}\right)\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}c\left(a_{11}{b}_{12}+a_{12}{b}_{22}\right)\\ c\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\left(a_{21}{b}_{11}+a_{22}{b}_{21}\right)\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}c\left(a_{21}{b}_{12}+a_{22}{b}_{22}\right)\end{array}\right]=$ تعريف الضرب في عدد ثابت.

$\left[\begin{array}{cc}\begin{array}{ll}c{a}_{11}{b}_{11}+c{a}_{12}{b}_{21}& c{a}_{11}{b}_{12}+c{a}_{12}{b}_{22}\\ c{a}_{21}{b}_{11}+c{a}_{22}{b}_{21}& c{a}_{21}{b}_{12}+c{a}_{22}{b}_{22}\end{array}& \end{array}\right]=$ خاصية التوزيع.

$\left[\begin{array}{cc}\begin{array}{ll}c{a}_{11}& c{a}_{12}\\ c{a}_{21}& c{a}_{22}\end{array}& \end{array}\right]\left[\begin{array}{cc}\begin{array}{ll}{b}_{11}& b_{12}\\ b_{21}& b_{22}\end{array}& \end{array}\right]=$ تعريف ضرب المصفوفات.

$c\left[\begin{array}{cc}\begin{array}{ll}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}& \end{array}\right]\left[\begin{array}{cc}\begin{array}{ll}{b}_{11}& b_{12}\\ b_{21}& b_{22}\end{array}& \end{array}\right]=$ تعريف الضرب في عدد ثابت.

$\left(c\underset{\_}{A}\right)\underset{\_}{B}=$ بالتعويض.

وبالمثل $c\left(\underset{\_}{A}\underset{\_}{B}\right)=\underset{\_}{A}\left(c\underset{\_}{B}\right)$

45) مسألة مفتوحة: اكتب مصفوفتين A وB على أن تكون BA = AB.

$\begin{array}{rl}\left[\begin{array}{cc}\begin{array}{ll}2& 2\\ 2& 2\end{array}& \end{array}\right]& =\underset{\_}{A}\\ \left[\begin{array}{cc}\begin{array}{ll}3& 3\\ 3& 3\end{array}& \end{array}\right]& =B\end{array}$

46) تحد: جد قيم d, c, b, a التي تجعل العبارة $\left[\begin{array}{cc}\begin{array}{ll}a& b\\ c& d\end{array}& \end{array}\right]\cdot \left[\begin{array}{cc}\begin{array}{ll}4& 3\\ 2& 5\end{array}& \end{array}\right]=\left[\begin{array}{cc}\begin{array}{ll}10& 11\\ 20& 29\end{array}& \end{array}\right]$ صحيحة.

$\begin{array}{rl}& 4a+2b=10\\ & 3a+5b=11\end{array}$

بضرب المعادلة الأولى في 5 والمعادلة الثانية في 2 وبالطرح.

$\begin{array}{rl}20a+10b& =50\\ -6a-10b& =-22\\ 14a& =28\end{array}$ بقسمة الطرفين على 14.

$\begin{array}{r}a=2\\ 4\left(2\right)+2b=10\\ 2b=2\\ b=1\end{array}$ بالتعويض عن قيمة a في المعادلة الأولى.

$\begin{array}{rl}& 4c+2d=20\\ & 3c+5d=29\end{array}$

بضرب المعادلة الأولى في 5 والمعادلة الثانية في 2 وبالطرح.

$\begin{array}{rl}20c+10d& =100\\ -6c-10d& =-58\\ 14c& =42\end{array}$ بقسمة الطرفين على 14.

$\begin{array}{rl}c=3& \\ 4\left(3\right)+2d& =20\\ 2d& =8\\ d& =4\end{array}$ بالتعويض عن قيمة c في المعادلة الأولى.

48) في مشغل ثلاث آلات حياكة فإذا كان إنتاج كل آلة في 3 أشهر كما في الجدول أدناه وسعر بيع القطعة الواحدة من إنتاج كل آلة معطى أيضاً فما المبلغ الذي سيحصل عليه المشغل من مبيعات الأشهر الثالثة؟

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49) رتبة المصفوفة: ما رتبة المصفوفة الناتجة عن عملية الضرب الآتية؟

$\left(\begin{array}{cc}\begin{array}{lll}a& b& c\\ d& e& f\\ g& h& i\\ j& k& l\end{array}& \end{array}\right)\cdot \left[\begin{array}{c}\begin{array}{l}7\\ 4\\ 6\end{array}\\ \end{array}\right]$

• 4×1
• 3×3
• 1×4
• 3×4