mathematics-class-12-unit-03-chapter-05-matrices-l-5-6-at3cnafn6xe

### <Success style="font-size:25px"> Matrices L-5 </Success>
### <primary style="font-size:24px"> System of unique solution </primary>
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- $-3 x-2 y+4 y=9$

- $\quad$$\quad$$\quad$$3 y-2 z=5$

- $\quad$$4 x-3 y+2 z=7$

- ${\left[\begin{array}{ccc}-3 & -2 & 4 \\\ 0 & 3 & -2 \\\ 4 & -3 & 2\end{array}\right]\left[\begin{array}{l}x \\\ y \\\ z\end{array}\right]=\left[\begin{array}{l}9 \\\ 5 \\\ 7\end{array}\right]}$

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<footer style = "font-size: 18px">System of linear equations $\rightarrow$ 3 possibilities of solutions $\rightarrow$ <span style = "color:blue"> System of unique solution </span> $\rightarrow$ System of unique solution contd... $\rightarrow$ System of unique solution contd... </footer>

### <Success style="font-size:25px"> Matrices L-5 </Success>
### <primary style="font-size:24px"> System of unique solution contd... </primary>
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- $\begin{aligned} & {\left[\begin{array}{ccc|c}-3 & -2 & 4 & 9 \\\ 0 & 3 & -2 & 5 \\\ 4 & -3 & 2 & 7\end{array}\right]} \\\ & R_1 \rightarrow \frac{1}{-3} R_1\left[\begin{array}{ccc|c}1 & \frac {2}{3} & \frac {-4}{3} & -3 \\\ 0 & 3 & -2 & 5 \\\ 4 & -3 & 2 & 7\end{array}\right] \\\ & R_3 \rightarrow R_3-4 R_1{\left[\begin{array}{ccc|c}1 & \frac {2}{3} & {-4}{3} & -3 \\\ 0 & 3 & -2 & 5 \\\ 0 & -3\frac {-2}{3} & 2+\frac{16}{3} & 7+12 \end{array}\right]}\\\ \end{aligned}$
	
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<footer style = "font-size: 18px">3 possibilities of solutions $\rightarrow$ System of unique solution $\rightarrow$ <span style = "color:blue"> System of unique solution contd... </span> $\rightarrow$ System of unique solution contd... $\rightarrow$ System of unique solution contd... </footer>

### <Success style="font-size:25px"> Matrices L-5 </Success>
### <primary style="font-size:24px"> System of unique solution contd... </primary>
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- $\begin{aligned} & {\left[\begin{array}{ccc|c}1 & \frac{2}{3} & -\frac{4}{3} & -3 \\\ 0 & 3 & -2 & 5 \\\ 0 & -\frac{17}{3} & \frac {22}{3} & 19\end{array}\right]} \\\ & R_2 \rightarrow \frac{1}{3} R_2\left[\begin{array}{ccc|c}1 & \frac{2}{3} & -\frac{4}{3} & -3 \\\ 0 & 1 & -\frac{2}{3} & \frac{5}{3} \\\ 0 & -\frac {17}{3} & \frac {22}{3} & 19\end{array}\right] \\\ & \begin{array}{l}R_1 \rightarrow R_1-\frac {2}{3} R_2 \\\ R_3 \rightarrow R_3+\frac{17}{3} R_2\end{array}\left[\begin{array}{ccc|c}1 & 0 & -\frac{4}{3}+\frac{4}{9} & -3- \frac{10}{3} \\\ 0 & 1 & -\frac{2}{3} & \frac{5}{3} \\\ 0 & 0 & \frac{22}{3}-\frac{34}{9} & 19+\frac{85}{9}\end{array}\right] \\\ & \end{aligned}$

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<footer style = "font-size: 18px">System of unique solution $\rightarrow$ System of unique solution contd... $\rightarrow$ <span style = "color:blue"> System of unique solution contd... </span> $\rightarrow$ System of unique solution contd... $\rightarrow$ System of unique solution contd... </footer>

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### <primary style="font-size:24px"> System of unique solution contd... </primary>
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- $\begin{aligned} & {\left[\begin{array}{ccc|c}1 & 0 & -\frac{8}{9} & -\frac{19}{3} \\\ 0 & 1 & -\frac{2}{3} & \frac{5}{3} \\\ 0 & 0 & \frac{32}{9} & \frac{256}{9}\end{array}\right]} \\\ & R_3 \rightarrow \frac{9}{32} R_3\left[\begin{array}{ccc|c}1 & 0 & -\frac{2}{9} & -\frac{19}{3} \\\ 0 & 1 & -\frac{2}{3} & \frac{5}{3} \\\ 0 & 0 & 1 & \frac{256}{32}\end{array}\right] \\\ & \begin{array}{l}R_1 \longrightarrow \frac{8}{9} R_3+R_1 \\\ R_2 \rightarrow \frac{2}{3} R_3+R_2\end{array}\left[\begin{array}{lll|l}1 & 0 & 0 & 3 \\\ 0 & 1 & 0 & 7 \\\ 0 & 0 & 1 & 8\end{array}\right] \\\ & \end{aligned}$

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<footer style = "font-size: 18px">System of unique solution contd... $\rightarrow$ System of unique solution contd... $\rightarrow$ <span style = "color:blue"> System of unique solution contd... </span> $\rightarrow$ System of unique solution contd... $\rightarrow$ Elementary row operations and system of linear equations</footer>

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<h3 id="primary-stylefont-size24px-elementary-row-operations-and-system-of-linear-equations-primary"><primary style="font-size:24px"> Elementary row operations and system of linear equations </primary></h3>
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<p>$A x=B$ be the given system of linear equations</p>
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<p>Let $\rho$ be the elementary row operations that one performed on $A$ & $B$. Let $\tilde{A}$ denote $\rho(A)$ & $\tilde{B}$ denote $\rho(B)$.</p>
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<footer style = "font-size: 18px">System of unique solution contd... $\rightarrow$ System of unique solution contd... $\rightarrow$ <span style = "color:blue"> Elementary row operations and system of linear equations </span> $\rightarrow$ Elementary row operations and system of linear equations contd... $\rightarrow$ Elementary row operations and system of linear equations contd... </footer>

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<p>$\tilde{A} x=\tilde{B}$ - the newly obtained system.</p>
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<p>$A$ is row equivalent to $\tilde{A}$ &</p>
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<p>$B$ is now equivalent to $\tilde{B}$</p>
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<p>i.e., $\vec{A}$ is obtained  from $A$ just by applying the row elementary operations.</p>
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<p>The systems,</p>
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<p>$A x=B$ & $ \tilde{A} x=\tilde{B} $</p>
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<p>have the same set of solutions.</p>
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<p>If $\rho$ is an elementary row operation then $\rho(A)=\rho(I) A$.</p>
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<footer style = "font-size: 18px">System of unique solution contd... $\rightarrow$ Elementary row operations and system of linear equations $\rightarrow$ <span style = "color:blue"> Elementary row operations and system of linear equations contd... </span> $\rightarrow$ Elementary row operations and system of linear equations contd... $\rightarrow$ Elementary row operations and system of linear equations contd... </footer>

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<p>Let, $\rho_1, \rho_2, \ldots, \rho_s$ be a finite set of elementary row operations. Let, $A$ be an $n \times n$ matrix.</p>
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<p>Then,</p>
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<p>a) $(\rho_s \circ \rho_{s-1} \circ \rho_{s-3} \circ \cdots \circ \rho_2 \circ \rho_1)$ (A)</p>
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<p>$=\left(\rho_s \circ \rho_{s-1} \cdot \cdots \circ \rho_2 \circ \rho_s\right)(I)(A)$</p>
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<p>b) $ \left(\rho_s \circ \rho_{s-1} \circ \cdots \circ \rho_2 \circ \rho_1\right)(A) $</p>
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<p>$ =\rho_s(I) \rho_{s-1}(I) \cdots \rho_2(I) \rho_1(I) A$</p>
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<p>Note:- A similar version also holds for $n \times m$ matrices also.</p>
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<footer style = "font-size: 18px">Elementary row operations and system of linear equations $\rightarrow$ Elementary row operations and system of linear equations contd... $\rightarrow$ <span style = "color:blue"> Elementary row operations and system of linear equations contd... </span> $\rightarrow$ Elementary row operations and system of linear equations contd... $\rightarrow$ Elementary row operations and system of linear equations contd... </footer>

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### <primary style="font-size:24px"> Elementary row operations and system of linear equations contd... </primary>
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- c) If $\rho$ is an elementary row operation & if $A$ & $B$ are any two matrices which can be multiplied, then

- $\rho(A B)  = \rho(A) \cdot B $

- $\rho(A B)  = \rho(I)(A B)$

- $ =(\rho(I) A)(B) $

- $ =\rho(A)(B) .$

- $\therefore$  If $\rho_s, \rho_{s-1}, \ldots . \rho_2, \rho_1$ is a finite set of elementary row operations then

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<footer style = "font-size: 18px">Elementary row operations and system of linear equations contd... $\rightarrow$ Elementary row operations and system of linear equations contd... $\rightarrow$ <span style = "color:blue"> Elementary row operations and system of linear equations contd... </span> $\rightarrow$ Elementary row operations and system of linear equations contd... $\rightarrow$ Elementary row operations and system of linear equations contd... </footer>

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<p>$ \left(\rho_{s} \circ \rho_{s-1} \circ \rho_{s-2} \circ \cdots \circ \rho_2 \circ \rho_1\right)(A B)$</p>
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<p>$ =\left(\rho_s \cdot \rho_{s-1} \cdot \rho_{s-2} \circ \cdots \circ \rho_2 \circ \rho_1\right)(A) B .$</p>
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<p>$A X=B$ is the given system.</p>
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<p>Suppose, $x$ is the solution to the system, then $\rho(Ax)=\rho(B)$</p>
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<footer style = "font-size: 18px">Elementary row operations and system of linear equations contd... $\rightarrow$ Elementary row operations and system of linear equations contd... $\rightarrow$ <span style = "color:blue"> Elementary row operations and system of linear equations contd... </span> $\rightarrow$ Elementary row operations and system of linear equations contd... $\rightarrow$ Elementary row operations and system of linear equations contd... </footer>

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<p>where, $\rho$ is any elementary row operation.</p>
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<p>$\rho(A) x=\rho(B) $</p>
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<p>$\tilde{A} x=\tilde{D}$</p>
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<footer style = "font-size: 18px">Elementary row operations and system of linear equations contd... $\rightarrow$ Elementary row operations and system of linear equations contd... $\rightarrow$ <span style = "color:blue"> Elementary row operations and system of linear equations contd... </span> $\rightarrow$ Elementary row operations and system of linear equations contd... $\rightarrow$ Over determined system </footer>

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<li>Thus, the systems $A X=B$ and $\tilde{A} X=\tilde{B}$ have the same set of solution if $\tilde{A}$ & $\tilde{B}$ is obtained from $A$ & $B$, respectively, just by applying a finite set of elementary row operations.</li>
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<footer style = "font-size: 18px">Elementary row operations and system of linear equations contd... $\rightarrow$ Elementary row operations and system of linear equations contd... $\rightarrow$ <span style = "color:blue"> Elementary row operations and system of linear equations contd... </span> $\rightarrow$ Over determined system $\rightarrow$ Solution of the over determined system </footer>

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<p>$ 2 x-3 y=-21$</p>
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<p>$ 3 x+2 y=1 $</p>
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<p>$ 8 x-5 y=-49$</p>
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<p>This is an over determined system</p>
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<footer style = "font-size: 18px">Elementary row operations and system of linear equations contd... $\rightarrow$ Elementary row operations and system of linear equations contd... $\rightarrow$ <span style = "color:blue"> Over determined system </span> $\rightarrow$ Solution of the over determined system $\rightarrow$ Solution of the over determined system contd... </footer>

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<p>${\left[\begin{array}{cc}2 & -3 \\ 3 & 2 \\ 8 & -5\end{array}\right]\left[\begin{array}{c}x \\ y\end{array}\right]=\left[\begin{array}{c}-21 \\ 1 \\ -49\end{array}\right] } $</p>
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<p>$ {\left[\begin{array}{cc|c}2 & -3 & -21 \\ 3 & 2 & 1 \\ 8 & -5 & -49\end{array}\right]}$</p>
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<p>$ R_1 \rightarrow \frac{1}{2} R_1\left[\begin{array}{cc|c}1 & -\frac{3}{2} & -\frac{21}{2} \\ 3 & 2 & 1 \\ 8 & -5 & -49\end{array}\right]$</p>
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<p>$\begin{aligned} & R_2 \rightarrow R_2-3 R_1 \\ & R_3 \rightarrow R_3-3 R_1\end{aligned}\left[\begin{array}{cc|c}1 & -\frac{3}{2} & -\frac{21}{2} \\ 0 & 2+\frac{9}{2} & 1+\frac{63}{2} \\ 0 & -5+\frac{24}{2} & -49+\frac{162}{2}\end{array}\right]$</p>
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<p>$= \left[\begin{array}{cc|c}1 & -\frac{3}{2}  & -\frac{21}{2} \\ 0 & \frac{12}{3}  & \frac{65}{2} \\ 0 &  \frac{14}{2} &  35\end{array}\right] $</p>
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<p>$ R_2 \rightarrow \frac{2}{13} R_2\left[\begin{array}{cc|c}1 & -\frac{3}{2} & -\frac{21}{2} \\ 0 & 1 & 5 \\ 0 & \frac{14}{2} & 35\end{array}\right] $</p>
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<footer style = "font-size: 18px">Over determined system  $\rightarrow$ Solution of the over determined system. $\rightarrow$ <span style = "color:blue"> Solution of the over determined system contd... </span> $\rightarrow$ Solution of the over determined system contd... $\rightarrow$ Problem </footer>

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<p>$\begin{aligned} & R_1 \rightarrow \frac{3}{2} R_2+R_1 \\ & R_3 \rightarrow \frac{-14}{2} R_2+R_3\end{aligned}\left[\begin{array}{ll|l}1 & 0 & \frac{15}{2}-\frac{21}{2} \\ 0 & 1 &\hspace {0.5cm} 5 \\ 0 & 0 & 35-35\end{array}\right]$</p>
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<p>$=\left[\begin{array}{cc|c}1 & 0 & -3 \\ 0 & 1 & 5 \\ 0 & 0 & 0\end{array}\right]$</p>
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<p>$x = -3, y = 5$</p>
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<p>Note:- A system of linear equations has a solution if $ \operatorname{rank}(A)=\operatorname{rank}(A \mid B)$.</p>
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<li>$\begin{aligned} & {\left[\begin{array}{ccc}2 & -3 & 2 \\ 1 & 1 & -1 \\ 3 & -4 & -3\end{array}\right]\left[\begin{array}{l}x \\ y \\ z\end{array}\right]=\left[\begin{array}{l}13 \\ 2 \\ 1\end{array}\right]} \\ & {\left[\begin{array}{ccc|c}2 & -3 & 2 & 13 \\ 3 & 1 & -1 & 2 \\ 3 & -4 & -3 & 1\end{array}\right]} \\ & R_1 \rightarrow \frac{1}{2} {R_1}\left[\begin{array}{ccc|c}1 & -\frac{3}{2} & 1 & \frac{13}{2} \\ 3 & 1 & -1 & 2 &  \\ 3 & -4 & -3 & 1\end{array}\right]\end{aligned}$</li>
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<p>$ {\left[\begin{array}{lll|l}1 & 0 & -\frac{1}{11} & \frac{38}{22} \\ 0 & 1 & -\frac{8}{11} & -\frac{35}{11} \\ 0 & 0 & -\frac{62}{11} & -\frac{372}{22}\end{array}\right]} $</p>
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<p>$ R_3 \rightarrow \frac{-11}{62} R_3\left[\begin{array}{ccc|c}1 & 0 & -\frac{1}{11} & \frac{38}{22} \\ 0 & 1 & -\frac{8}{11} & -3 \frac{5}{11} \\ 0 & 0 & 1 & 3\end{array}\right] $</p>
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<p>$ \begin{array}{l}R_1 \longrightarrow R_1+\frac{1}{11} R_3 ,\quad R_2 \longrightarrow R_2+\frac{8}{11} R_3\end{array}$</p>
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<p>$ \left[\begin{array}{lll|l}1 & 0 & 0 & \frac{38}{22}+\frac{3}{11} \\ 0 & 1 & 0 & -\frac{36}{11}+\frac{24}{11} \\ 0 & 0 & 1 & 3\end{array}\right] $</p>
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<p>$= {\left[\begin{array}{lll|l}1 & 0 & 0 & 2 \\ 0 & 1 & 0 & -1 \\ 0 & 0 & 1 & 3\end{array}\right]} $</p>
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<p>$ (2,-1,3) $</p>
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