تمارين (1-3)

(1)- جد قيمتي x, y حيث $\mathrm{x},\mathrm{y}\in \mathrm{R}$ في كل مما يأتي:

أ- $\left[\begin{array}{cl}3\mathrm{x}+\mathrm{y}& 0.2\\ 3\sqrt{2}& \mathrm{x}-\mathrm{y}\end{array}\right]=\left[\begin{array}{ll}8& \frac{1}{5}\\ 3\sqrt{2}& 0\end{array}\right]$

بما أن المصفوفتان متساويتان فالعناصر المتناظرة فيهما متساويتين.

$$\begin{gathered} \begin{array}{rl}& 3x+y=8\\ & x-y=0\end{array} \\ 2x=8\to x=\frac{8}{2}=4 \\ \begin{array}{rl}& (3\times 2)+\mathrm{y}=8\\ & \mathrm{y}=8-6\to \mathrm{y}=2\in \mathrm{R}\end{array} \end{gathered}$$

ب- $\left[\begin{array}{cc}\sin \mathrm{x}& 3\\ -2& \cos \mathrm{x}\end{array}\right]=\left[\begin{array}{cc}0.5& 3\\ -2& \frac{1}{\sqrt{2}}\end{array}\right]$

بما أن المصفوفتان متساويتان فالعناصر المتناظرة فيهما متساويتين.

$\sin \mathrm{x}=0.5=\frac{1}{2}\to \mathrm{x}={30}^{\circ }, \cos \mathrm{x}=\frac{1}{\sqrt{2}}\leftarrow \mathrm{x}={45}^{\circ }$

ج- $\left[\begin{array}{cc}{\mathrm{x}}^2& 6\\ {\mathrm{y}}^2-\mathrm{y}& 4\end{array}\right]=\left[\begin{array}{cc}9& 6\\ 15& 4\end{array}\right]$

بما أن المصفوفتان متساويتان فالعناصر المتناظرة فيهما متساويتين.

$$\begin{gathered} \begin{array}{rl}& {\mathrm{x}}^2=9\to \mathrm{x}=\mp 3\\ & {\mathrm{y}}^2-2\mathrm{y}=15\to {\mathrm{y}}^2-2\mathrm{y}-15=0\\ & (\mathrm{y}-5)(\mathrm{y}+3)=0\end{array} \\ \begin{array}{rl}& (\mathrm{y}-5)=0\to \mathrm{y}=5\\ & \mathrm{or} (\mathrm{y}+3)=0\to \mathrm{y}=-3\end{array} \\ \mathrm{x}=\{3,-3\} \end{gathered}$$

(2)- جد ناتج ما يلي:

أ- $\left[\begin{array}{ccc}3& 1& 0\\ 5& -2& 1\end{array}\right]+\left[\begin{array}{ccc}-2& 4& 6\\ -11& 2& 3\end{array}\right]$

$\left[\begin{array}{ccc}3+(-2)& 1+4& 0+6\\ 5+(-11)& (-2)+2& 1+3\end{array}\right]=\left[\begin{array}{ccc}1& 5& 6\\ -6& 0& 4\end{array}\right]$

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

$$\begin{gathered} \left[\begin{array}{cc}\frac{1}{2}\times 4& -\frac{1}{4}\times 4\\ 2\times 4& -1\times 4\end{array}\right]+\left[\begin{array}{cc}-2\times 3& 1\times 3\\ 0\times 3& 3\times 3\end{array}\right] \\ =\left[\begin{array}{ll}2& -1\\ 8& -4\end{array}\right]+\left[\begin{array}{cc}-6& 3\\ 0& 9\end{array}\right]=\left[\begin{array}{cc}2+(-6)& -1+3\\ 8+0& -4+9\end{array}\right]=\left[\begin{array}{cc}-4& 2\\ 8& 5\end{array}\right] \end{gathered}$$

ج- $\left[\begin{array}{cc}3\sqrt{5}& -\frac{1}{4}\\ \sqrt{2}& -\sqrt{3}\\ 2& 5\end{array}\right]+\left[\begin{array}{cc}-\sqrt{5}& \frac{1}{8}\\ \sqrt{2}& 3\sqrt{3}\\ 3& -2\end{array}\right]$

$\left[\begin{array}{cc}3\sqrt{5}+(-\sqrt{5})& \frac{1}{4}\\ \sqrt{2}+\sqrt{2}& -\sqrt{3}+3\sqrt{3}\\ 2+3& 5+(-2)\end{array}\right]=\left[\begin{array}{cc}2\sqrt{5}& -\frac{1}{8}\\ 2\sqrt{2}& 2\sqrt{3}\\ 5& 3\end{array}\right]$

د- $\left[\begin{array}{c}3\\ -1\\ 1.6\end{array}\right]+2\left[\begin{array}{c}1\\ -2\\ 0.9\end{array}\right]+\left[\begin{array}{c}-0.4\\ 3\\ 2\end{array}\right]$

$\left[\begin{array}{c}3\\ 1\\ 1.6\end{array}\right]+\left[\begin{array}{c}2\\ -4\\ 1.8\end{array}\right]+\left[\begin{array}{c}-0.4\\ 3\\ 2\end{array}\right]=\left[\begin{array}{c}4.6\\ -2\\ 5.4\end{array}\right]$

(3)- جد المصفوفة x في كل مما يأتي:

أ- $2\mathrm{x}+\left[\begin{array}{rr}5& -2\\ 4& 1\end{array}\right]=\left[\begin{array}{rr}8& 7\\ -5& 3\end{array}\right]$

$$\begin{gathered} \begin{array}{rl}& 2\mathrm{x}+\left[\begin{array}{cc}5& 2\\ 4& 1\end{array}\right]=\left[\begin{array}{cc}5& -2\\ 4& 1\end{array}\right]=\left[\begin{array}{cc}8& 7\\ -5& 3\end{array}\right]-\left[\begin{array}{cc}5& -2\\ 4& 1\end{array}\right]\\ & 2\mathrm{x}=\left[\begin{array}{cc}8& 7\\ -5& 3\end{array}\right]+\left[\begin{array}{cc}-5& 2\\ -4& -1\end{array}\right]=\left[\begin{array}{cc}8+(-5)& 7+2\\ -5+(-4)& 3+(-1)\end{array}\right]\end{array} \\ 2\mathrm{x}=\left[\begin{array}{cc}3& 9\\ -9& 2\end{array}\right]\to \mathrm{x}=\left[\begin{array}{cc}\frac{3}{2}& \frac{9}{2}\\ \frac{-9}{2}& 1\end{array}\right] \end{gathered}$$

ب- $\mathrm{x}-\left[\begin{array}{ccc}3& 4& -1\\ 5& 2& -3\\ 6& -1& 7\end{array}\right]+\left[\begin{array}{ccc}-2& 7& 5\\ 6& 8& -2\\ -4& 3& 2\end{array}\right]=\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]$

$$\begin{gathered} \mathrm{x}+\left[\begin{array}{ccc}-2& 7& 5\\ 6& 8& -2\\ -4& 3& 2\end{array}\right]-\left[\begin{array}{ccc}3& 4& -1\\ 5& 2& -3\\ 6& -1& 7\end{array}\right]=\left[\begin{array}{lll}0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right] \\ \mathrm{x}+\left[\begin{array}{ccc}-5& 3& 6\\ 1& 6& 1\\ -10& 4& -5\end{array}\right]=\left[\begin{array}{lll}0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right] \\ \mathrm{x}+\left[\begin{array}{ccc}-5& 3& 6\\ 1& 6& 1\\ -10& 4& -5\end{array}\right]-\left[\begin{array}{ccc}-5& 3& 6\\ 1& 6& 1\\ -10& 4& -5\end{array}\right]=\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]-\left[\begin{array}{ccc}-5& 3& 6\\ 1& 6& 1\\ -10& 4& -5\end{array}\right] \\ \mathrm{x}=-\left[\begin{array}{ccc}-5& 3& 6\\ 1& 6& 1\\ -10& 4& -5\end{array}\right]\to \mathrm{x}=\left[\begin{array}{ccc}5& -3& -6\\ -1& -6& -1\\ 10& -4& 5\end{array}\right] \end{gathered}$$

(4)- إذا كانت [2 7 0] = A = [1 2 5], B = [- 4 3 -2], C جد المصفوفات الآتية:

أ- $2\mathrm{A}+3\mathrm{B}+\mathrm{C}$

$\begin{array}{rl}& 2\mathrm{A}=2\times \left[\begin{array}{lll}1& -2& 5\end{array}\right]=\left[\begin{array}{lll}2& -4& 10\end{array}\right]\\ & 3\mathrm{B}=3\times \left[\begin{array}{lll}-4& 3& -2\end{array}\right]=\left[\begin{array}{lll}-12& 9& -6\end{array}\right] , \mathrm{C}=\left[\begin{array}{lll}0& -7& 2\end{array}\right]\\ & 2\mathrm{A}+3\mathrm{B}+\mathrm{C}=\left[\begin{array}{lll}2& -4& 10\end{array}\right]+\left[\begin{array}{lll}-12& 9& -6\end{array}\right]+\left[\begin{array}{lll}0& -7& 2\end{array}\right]\\ & 2\mathrm{A}+3\mathrm{B}+\mathrm{C}=\left[\begin{array}{lll}-10& 12& 6\end{array}\right]\end{array}$

ب- $\mathrm{A}-\mathrm{B}+5\mathrm{C}$

$\begin{array}{rl}& \mathrm{A}=\left[\begin{array}{lll}1& -2& 5\end{array}\right],-\mathrm{B}=\left[\begin{array}{lll}4& -3& 2\end{array}\right]\\ & 5\mathrm{C}=5\times \left[\begin{array}{lll}0& 7& 2\end{array}\right]=\left[\begin{array}{lll}0& 35& 10\end{array}\right]\\ & \mathrm{A}-\mathrm{B}+5\mathrm{C}=\left[\begin{array}{lll}1+4+0& -2+(-3)+35& 5+2+10\end{array}\right]\\ & \mathrm{A}-\mathrm{B}+5\mathrm{C}=\left[\begin{array}{lll}5& 30& 17\end{array}\right]\end{array}$

ج- $3\mathrm{A}+\mathrm{B}+\mathrm{C}$

$\begin{array}{rl}& 3\mathrm{A}=3\times \left[\begin{array}{lll}1& -2& 5\end{array}\right]=\left[\begin{array}{lll}3& -6& 15\end{array}\right]\\ & \mathrm{B}=\left[\begin{array}{lll}-4& 3& -2\end{array}\right] , \mathrm{C}=\left[\begin{array}{lll}0& 7& 2\end{array}\right]\\ & 3\mathrm{A}+\mathrm{B}+\mathrm{C}=[3+(-4)+0-6+3+715+(-2)+2]\\ & 3\mathrm{A}+\mathrm{B}+\mathrm{C}=\left[\begin{array}{lll}-1& 4& 15\end{array}\right]\end{array}$

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

$$\begin{gathered} \begin{array}{rl}& -\mathrm{A}=-\left[\begin{array}{lll}1& -2& 5\end{array}\right]=\left[\begin{array}{lll}-1& 2& -5\end{array}\right]\\ & 2\mathrm{B}=2\times \left[\begin{array}{lll}-4& 3& -2\end{array}\right]=\left[\begin{array}{lll}-8& 6& -4\end{array}\right]\\ & -\mathrm{C}=-\left[\begin{array}{lll}0& 7& 2\end{array}\right]=\left[\begin{array}{lll}-0& -7& -2\end{array}\right]\end{array} \\ -\mathrm{A}+2\mathrm{B}-\mathrm{C}=\left[\right(-1)+(-8)+(-0)2+6+(-7\left)\right(-5)+(-4)+(-2\left)\right] \\ -\mathrm{A}+2\mathrm{B}-\mathrm{C}=\left[\begin{array}{lll}-9& 1& -11\end{array}\right] \end{gathered}$$