تمارين (1-9)

(1)- جد قيم x, y, z, h

أ- $\left[\begin{array}{cc}\mathrm{x}-2& 2\mathrm{y}+1\\ \mathrm{x}+3& 16\end{array}\right]=\left[\begin{array}{cr}3& -5\\ \mathrm{z}& 3\mathrm{h}-2\end{array}\right]$

$$\begin{gathered} \begin{array}{rl}& \mathrm{x}-2=3\to \mathrm{x}=5\\ & \mathrm{x}+3=\mathrm{z}\\ & 5+3=\mathrm{z}\\ & \mathrm{z}=8\end{array} \\ \begin{array}{rl}& 2\mathrm{y}+1=-5\\ & 2\mathrm{y}=-6\\ & \mathrm{y}=-3\\ & 3\mathrm{h}-2=16\\ & 3\mathrm{h}=16+2\\ & 3\mathrm{h}=18\\ & \mathrm{h}=6\end{array} \end{gathered}$$

$\mathrm{x}=5\mathrm{z}=8\mathrm{h}=6\mathrm{y}=-3\left[\begin{array}{ll}3& -5\\ 8& 16\end{array}\right]=\left[\begin{array}{ll}3& -5\\ 8& 16\end{array}\right]$

ب- $\left[\begin{array}{rr}3\mathrm{x}& 10\\ 2\mathrm{x}+\mathrm{z}& 2\mathrm{y}-\mathrm{h}\end{array}\right]=\left[\begin{array}{cc}15& 2\mathrm{y}\\ 10& 0\end{array}\right]$

$$\begin{gathered} \begin{array}{rl}& 3\mathrm{x}=15\to \mathrm{x}=5\\ & 2\mathrm{y}-\mathrm{h}=0\\ & 2\times 5-\mathrm{h}=0\\ & \mathrm{h}=10\end{array} \\ \begin{array}{rl}& 2\mathrm{y}=10\to \mathrm{y}=5\\ & 2\mathrm{x}+\mathrm{z}=10\\ & 2\times 5+\mathrm{z}=10\\ & 10+\mathrm{z}=10\\ & \mathrm{z}=10-10\to \mathrm{z}=0\end{array} \end{gathered}$$

$\mathrm{x}=5\mathrm{z}=0\mathrm{y}=5\mathrm{h}=10\left[\begin{array}{cc}15& 10\\ 10& 0\end{array}\right]=\left[\begin{array}{cc}15& 10\\ 10& 0\end{array}\right]$

(2)- أجر العمليات الآتية إن أمكن، مع ذكر السبب في حالة تعذر إجراء العملية:

أ- $\left[\begin{array}{ccc}3& 2& 0\\ 1& 2& -4\end{array}\right]+\left[\begin{array}{ccc}0& -1& 2\\ 3& 4& -5\end{array}\right]$

$\left[\begin{array}{ccc}3& 2& 0\\ 1& 2& -4\end{array}\right]+\left[\begin{array}{ccc}0& -1& 2\\ 3& 6& -5\end{array}\right]=\left[\begin{array}{ccc}3& 1& 2\\ 4& 6& -9\end{array}\right]$

ب- $\left[\begin{array}{ccc}3& -1& 2\\ 4& 5& 6\end{array}\right]+\left[\begin{array}{rr}0& 4\\ 5& -6\end{array}\right]$

غير ممكن لأنهما من نوعين مختلفين وليس لهما نفس القياس والحجم.

ج- $\left[\begin{array}{ll}6& 8\end{array}\right]+\left[\begin{array}{l}2\\ 1\end{array}\right]$

غير ممكن لأنهما مختلفان فالأولى صف والثانية عمود.

د- $\left[\begin{array}{ll}\mathrm{a}& \mathrm{b}\\ \mathrm{c}& \mathrm{d}\end{array}\right]-\left[\begin{array}{ll}\mathrm{x}& \mathrm{y}\end{array}\right]$

غير ممكن لأنهما من قياسين مختلفين.

(3)- إذا كانت $\mathrm{A}=\left[\begin{array}{cc}-2& -3\\ 1& 2\\ 0& 5\end{array}\right]$ فجد المصفوفة k, A عندما تكون:

أ- $\mathrm{k}=1$

$\mathrm{k}\cdot \mathrm{A}=1\times \left[\begin{array}{cc}-2& -3\\ 1& 2\\ 0& 5\end{array}\right]=\left[\begin{array}{cc}-2& -3\\ 1& 2\\ 0& 5\end{array}\right]$

ب- $\mathrm{k}=\frac{2}{5}$

$\mathrm{k}\cdot \mathrm{A}=\frac{2}{5}\left[\begin{array}{cc}-2& -3\\ 1& 2\\ 0& 5\end{array}\right]=\left[\begin{array}{cc}\frac{-4}{5}& -\frac{6}{5}\\ \frac{2}{5}& \frac{4}{5}\\ 0& 2\end{array}\right]$

ج- $\mathrm{k}=0$

$\mathrm{k}\cdot \mathrm{A}=0\times \left[\begin{array}{cc}-2& -3\\ -1& 2\\ 0& 5\end{array}\right]=\left[\begin{array}{ll}0& 0\\ 0& 0\\ 0& 0\end{array}\right]$

د- $\mathrm{k}=-1$

$\mathrm{k}\cdot \mathrm{A}=-1\times \left[\begin{array}{cc}-2& -3\\ 1& 2\\ 0& 5\end{array}\right]=\left[\begin{array}{cc}2& 3\\ -1& -2\\ 0& -5\end{array}\right]$

هـ- $\mathrm{k}=2$

$\mathrm{k}\cdot \mathrm{A}=2\times \left[\begin{array}{cc}-2& -3\\ 1& 2\\ 0& 5\end{array}\right]=\left[\begin{array}{cc}-4& -6\\ 2& 4\\ 0& 10\end{array}\right]$

(4)- إذا كانت $\mathrm{C}=\left[\begin{array}{ll}0& -5\end{array}\right] , \mathrm{B}=\left[\begin{array}{ll}0& 5\end{array}\right] , \mathrm{A}=\left[\begin{array}{ll}5& 1\end{array}\right]$فعبر عن كل مما يأتي كمصفوفة:

أ- $\mathrm{A}+(\mathrm{B}+\mathrm{C})$

$\begin{array}{rl}\mathrm{A}+(\mathrm{B}+\mathrm{C})& =\left[\begin{array}{ll}5& 1\end{array}\right]+\left[\begin{array}{ll}0& 5\end{array}\right]+\left[\begin{array}{ll}0& -5\end{array}\right]\\ & =\left[\begin{array}{ll}5& 1\end{array}\right]+\left[\begin{array}{ll}0& 0\end{array}\right]=\left[\begin{array}{ll}5& 1\end{array}\right]\end{array}$

ب- $2\mathrm{A}+\mathrm{B}+\mathrm{C}$

$\begin{array}{rl}2\mathrm{A}+\mathrm{B}-\mathrm{C}& =2\left[\begin{array}{ll}5& 1\end{array}\right]+\left[\begin{array}{ll}0& 5\end{array}\right]-\left[\begin{array}{ll}0& -5\end{array}\right]\\ & =\left[\begin{array}{ll}10& 2\end{array}\right]+\left[\begin{array}{ll}0& 5\end{array}\right]+\left[\begin{array}{ll}0& 5\end{array}\right]\\ & =\left[\begin{array}{ll}10& 2\end{array}\right]+\left[\begin{array}{ll}0& 10\end{array}\right]\\ & =\left[\begin{array}{ll}10& 12\end{array}\right]\end{array}$

(5)- باستعمال المصفوفات A ،B ،C الواردة في التمرين (4) حل كلاً من المعادلات المصفوفة الآتية:

أ- $\mathrm{A}+\mathrm{X}=\mathrm{B}+\mathrm{C}$

$\begin{array}{rl}\mathrm{x}& =\mathrm{B}+\mathrm{C}-\mathrm{A}\\ & =\left[\begin{array}{ll}0& 5\end{array}\right]+\left[\begin{array}{ll}0& -5\end{array}\right]-\left[\begin{array}{ll}5& 1\end{array}\right]\\ & =\left[\begin{array}{ll}0& 0\end{array}\right]+\left[\begin{array}{ll}-5& -1\end{array}\right]\\ & =\left[\begin{array}{ll}-5& -1\end{array}\right]\end{array}$

ب- $2(\mathrm{B}-\mathrm{C})=2(\mathrm{X}-\mathrm{C})-\mathrm{B}$

$\begin{array}{rl}& 2\mathrm{B}-2\mathrm{C}=2\mathrm{x}-2\mathrm{C}-\mathrm{B}\\ & 2\mathrm{B}+\mathrm{B}=2\mathrm{x}\\ & 2\mathrm{x}=3\mathrm{B}\to \mathrm{x}=\frac{3}{2}\mathrm{B}\\ & \mathrm{x}=\frac{3}{2}\mathrm{B}\to \mathrm{x}=\frac{3}{2}\left[\begin{array}{ll}0& 5\end{array}\right]=\left[\begin{array}{ll}0& \frac{15}{2}\end{array}\right]\end{array}$

ج- $\frac{1}{2}(\mathrm{A}+\mathrm{X})-3\mathrm{X}+2\mathrm{B}$

$$\begin{gathered} \begin{array}{rl}& \frac{1}{2}\mathrm{A}+\frac{1}{2}\mathrm{x}=3\mathrm{x}+2\mathrm{B}\\ & \frac{1}{2}\mathrm{A}-2\mathrm{B}=3\mathrm{x}-\frac{1}{2}\mathrm{x}\\ & \frac{1}{2}\mathrm{A}-2\mathrm{B}=2\frac{1}{2}\mathrm{x}\\ & 2\frac{1}{2}\mathrm{x}=\frac{1}{2}\mathrm{A}-2\mathrm{B}\\ & (\frac{5}{2}\mathrm{x}=\frac{1}{2}\mathrm{A}-2\mathrm{B})\frac{2}{5}\\ & \mathrm{x}=\frac{1}{5}\mathrm{A}-\frac{4}{5}\mathrm{B}\end{array} \\ \begin{array}{rl}\mathrm{x}& =\frac{1}{5}\mathrm{A}-\frac{4}{5}\mathrm{B}\\ & =\frac{1}{5}\left[\begin{array}{ll}5& 1\end{array}\right]-\frac{4}{5}\left[\begin{array}{ll}0& 5\end{array}\right]\\ & =\left[\begin{array}{ll}1& \frac{1}{5}\end{array}\right]-\left[\begin{array}{ll}0& 4\end{array}\right]\\ & =\left[\begin{array}{ll}1& \frac{1}{5}\end{array}\right]+\left[\begin{array}{ll}0& -4\end{array}\right]\\ & =\left[\begin{array}{ll}1& -3\frac{4}{5}\end{array}\right]\\ & =\left[\begin{array}{ll}1& -\frac{19}{5}\end{array}\right]\end{array} \end{gathered}$$