mathematics-class-12-unit-04-chapter-04-determinants-l-4-11-iyd-epc3cfs

### <Success style="font-size:25px"> Determinants L-4 </Success>
### <primary style="font-size:24px"> System of equations </primary>
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- Idea
 
- $ \underbrace{a}_ {\text{scalar}} ~ \underbrace{x}_ {\text{unknown}} = \underbrace{b}_ {\text{scalar}}$

-  x unknown that needs to be solved
 
- $a \neq 0, x=b / a$
 
-  generalize these
 
- $\begin{aligned} & a x+b y=m\end{aligned} ~, \begin{aligned} & c x+d y=n\end{aligned}$
 
- $\text {}\left[\begin{array}{ll}a & b \\\ c & d\end{array}\right]\left[\begin{array}{l}x \\\y \end{array}\right]=\left[\begin{array}{l}m \\\ n\end{array}\right]$  
 
-  are unknowns solutions?
 
- $A x=B \Rightarrow {x}=$ ?
 
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<footer style = "font-size: 18px"> $\rightarrow$ Determinants lecture - 4 $\rightarrow$ <span style = "color:blue"> System of equations </span> $\rightarrow$ System of equations $\rightarrow$ Example </footer>

### <Success style="font-size:25px"> Determinants L-4 </Success>
### <primary style="font-size:24px"> System of equations </primary>
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- $\text {}\left[\begin{array}{ll}a & b \\\ c & d\end{array}\right]\left[\begin{array}{l}x \\\y \end{array}\right]=\left[\begin{array}{l}m \\\ n\end{array}\right]$
 
-  Goal: How to solve for $x$ ?

-  Notation: $\underline{x} \equiv x$

-  (in appropriate context, we use ' $x$ ' to denote the vector value)
 
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<footer style = "font-size: 18px">Determinants <br> lecture - 4 $\rightarrow$ System of equations $\rightarrow$ <span style = "color:blue"> System of equations </span> $\rightarrow$ Example $\rightarrow$  Consistency and inconsistency of solution </footer>

### <Success style="font-size:25px"> Determinants L-4 </Success>
### <primary style="font-size:24px"> Example </primary>
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- Example:
 
- $\begin{aligned} & a_{11} x+a_{12} y+a_{13} z=b_1 \\end{aligned}$ $\longrightarrow$ (1)

- $\begin{aligned} & a_{21} x+a_{22} y+a_{23} z=b_2 \\end{aligned}$ $\longrightarrow$ (2)

- $\begin{aligned} & a_{31} x+a_{32} y+a_{33} z=b_3 \\end{aligned}$ $\longrightarrow$ (3)

- $\left[\begin{array}{lll}a_{11} & a_{12} & a_{13} \\\ a_{21} & a_{22} & a_{23} \\\ a_{31} & a_{32} & a_{33}\end{array}\right] \left[\begin{array}{l}x \\\ y \\\ z\end{array}\right]=\left[\begin{array}{l}b_1 \\\ b_2 \\\ b_3\end{array}\right]$
 
- $A \times x = B$
 
-  In general, $n \times n$($A$)   $n \times1$($x$)  $n \times1$($B$)
 
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<footer style = "font-size: 18px">System of equations $\rightarrow$ System of equations $\rightarrow$ <span style = "color:blue"> Example </span> $\rightarrow$  Consistency and inconsistency of solution $\rightarrow$  Consistency and inconsistency of solution case-1 </footer>

### <Success style="font-size:25px"> Determinants L-4 </Success>
### <primary style="font-size:24px">  Consistency and inconsistency of solution </primary>
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- $A \underline{x}=B \rightarrow \text { System of Equations }$

-  Consistency: A system of equations is said to be consistent if a solution exists (one or more solutions)

-  Inconsistency: A system of equations is said to be inconsistent if a solution does not exist
 
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<footer style = "font-size: 18px">System of equations $\rightarrow$ Example $\rightarrow$ <span style = "color:blue">  Consistency and inconsistency of solution </span> $\rightarrow$  Consistency and inconsistency of solution case-1 $\rightarrow$  Consistency and inconsistency of solution case-2 </footer>

### <Success style="font-size:25px"> Determinants L-4 </Success>
### <primary style="font-size:24px">  Consistency and inconsistency of solution case-1 </primary>
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- $A \underline{x}=B$
 
- consistency, inconsistency?

- How to check?

- Case 1: A is non. singular $(\operatorname{det}(A) \neq 0)$ $\Rightarrow A^{-1}$ exists

- $ A^{-1} \times [A \underline{x}=B $

- $ \Rightarrow \underbrace{{A}^{-1}{A}}  \underline{x}=A^{-1} B $
 
- $ \Rightarrow \underline{x}=A^{-1} B$
 
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<footer style = "font-size: 18px">Example $\rightarrow$  Consistency and inconsistency of solution $\rightarrow$ <span style = "color:blue">  Consistency and inconsistency of solution case-1 </span> $\rightarrow$  Consistency and inconsistency of solution case-2 $\rightarrow$  Consistency and inconsistency of solution case-2 </footer>

### <Success style="font-size:25px"> Determinants L-4 </Success>
### <primary style="font-size:24px">  Consistency and inconsistency of solution case-2 </primary>
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-  Case 2: $A$ is singular $(det(A)=0)$

- $ (adj A) A= det(A) \cdot I=0 $

- $ \rightarrow \times[A \underline{x}=B $
 
- $ \Rightarrow(\underbrace{adj A) A}_0 \underline{x}=(adj A) B $

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<footer style = "font-size: 18px"> Consistency and inconsistency of solution $\rightarrow$  Consistency and inconsistency of solution case-1 $\rightarrow$ <span style = "color:blue">  Consistency and inconsistency of solution case-2 </span> $\rightarrow$  Consistency and inconsistency of solution case-2 $\rightarrow$ Example-1 </footer>

### <Success style="font-size:25px"> Determinants L-4 </Success>
### <primary style="font-size:24px">  Consistency and inconsistency of solution case-2 </primary>
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- $ \Rightarrow 0 = (adj A) B$

-  (2a) $y(adj A) B=0$
 
-  then we cannot say anything about consistency or inconsistency$\rightarrow(2 b) 4(\operatorname{adj} A) B \neq 0$
 
-  then we say is 'inconsistent'

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<footer style = "font-size: 18px"> Consistency and inconsistency of solution case-1 $\rightarrow$  Consistency and inconsistency of solution case-2 $\rightarrow$ <span style = "color:blue">  Consistency and inconsistency of solution case-2 </span> $\rightarrow$ Example-1 $\rightarrow$ Example-1 </footer>

### <Success style="font-size:25px"> Determinants L-4 </Success>
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- $\text {}\\left[\begin{array}{l} 1 & 1 \\\ 4 & -1\end{array}\right] \text {}\left[\begin{array}{l}x \\\y \end{array}\right]=\text {}\left[\begin{array}{l}10 \\\ 0\end{array}\right]$

- $\ A \times x = B$
 
- $\\operatorname{det}(A)=-1-4=-5 \neq 0$

- $\Rightarrow$ This has a solution $\Rightarrow$ consistent

-  what is oration? $\underline{x}=A^{-1} B$
 
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<footer style = "font-size: 18px"> Consistency and inconsistency of solution case-2 $\rightarrow$  Consistency and inconsistency of solution case-2 $\rightarrow$ <span style = "color:blue"> Example-1 </span> $\rightarrow$ Example-1 $\rightarrow$ Example-2 </footer>

### <Success style="font-size:25px"> Determinants L-4 </Success>
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- $\ A^{-1}=\frac{-1}{5}\left[\begin{array}{cc}-1 & -1 \\\ -4 & 1\end{array}\right]$
 
- $\Rightarrow \ A^{-1} B=-\frac{1}{5}\left[\begin{array}{cc}-1 & -1 \\\ -4 & 1\end{array}\right]\left[\begin{array}{c}10 \\\ 0\end{array}\right]$
 
- $\Rightarrow \ \underline{x}=-\frac{1}{5}\left[\begin{array}{l}-10 \\\ -40\end{array}\right]=\left[\begin{array}{l}2 \\\ 8\end{array}\right]$
 
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<footer style = "font-size: 18px"> Consistency and inconsistency of solution case-2 $\rightarrow$ Example-1 $\rightarrow$ <span style = "color:blue"> Example-1 </span> $\rightarrow$ Example-2 $\rightarrow$ Example-2 </footer>

### <Success style="font-size:25px"> Determinants L-4 </Success>
### <primary style="font-size:24px"> Example-3 </primary>
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-  An inconsistent system?
 
- $\ x + y = 10 ~ \text{and}~ \ x + y = 20$
 
- $\ \text {}\left[\begin{array}{l} 1 & 1 \\\ 1 & 1\end{array}\right]\text {}\left[\begin{array}{l} x \\\y \end{array}\right]=\text {}\left[\begin{array}{l}10 \\\ 20 \end{array}\right]$
 
- $\ A \times x = B$
 
- $\ \operatorname{det}(A)=1-1=0$
 
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<footer style = "font-size: 18px">Example-2 $\rightarrow$ Example-2 $\rightarrow$ <span style = "color:blue"> Example-3 </span> $\rightarrow$ Example-3 $\rightarrow$ Example-3 </footer>

### <Success style="font-size:25px"> Determinants L-4 </Success>
### <primary style="font-size:24px"> Example-3 </primary>
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- $\ \operatorname{adj}(A)=\left[\begin{array}{cc}1 & -1 \\\ -1 & 1\end{array}\right] \times\left[\begin{array}{ll}A & \underline{x}\end{array}\right]=B$
 
-  check adj $\ (A) \cdot B=\left[\begin{array}{cc}1 & -1 \\\ -1 & 1\end{array}\right]\left[\begin{array}{l}10 \\\ 20\end{array}\right]=\left[\begin{array}{c}-10 \\\ 10\end{array}\right]$
 
- $\Rightarrow \ 0=\left[\begin{array}{c}-10 \\\ 10\end{array}\right]$?? $ \Rightarrow \text { inconsistent }$
 
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<footer style = "font-size: 18px">Example-2 $\rightarrow$ Example-3 $\rightarrow$ <span style = "color:blue"> Example-3 </span> $\rightarrow$ Example-3 $\rightarrow$ Example-3 </footer>

### <Success style="font-size:25px"> Determinants L-4 </Success>
### <primary style="font-size:24px"> Example-3 </primary>
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-  Infinitily many solutions?
 
- $ x+y  =10 $
 
- $ 2x+2y  =10 \times 2=20$
 
- $\text {}\left[\begin{array}{l} 1 & 1 \\\ 2 & 2\end{array}\right] \text {}\left[\begin{array}{l} x \\\y \end{array}\right]=\text {}\left[\begin{array}{l}10 \\\ 20 \end{array}\right]$
 
- $A \times x = B$
 
- $ det(A)=0 $ 
 
 
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<footer style = "font-size: 18px">Example-3 $\rightarrow$ Example-3 $\rightarrow$ <span style = "color:blue"> Example-3 </span> $\rightarrow$ Example-3 $\rightarrow$ Summary </footer>

### <Success style="font-size:25px"> Determinants L-4 </Success>
### <primary style="font-size:24px"> Example-3 </primary>
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- $\begin{aligned}\operatorname{adj}(A)=\left[\begin{array}{cc}2 & -1 \\\ -2 & 1\end{array}\right]\end{aligned}$
 
- $\begin{aligned}\operatorname{adj}(A). B =\left[\begin{array}{cc}0 \\\ 0\end{array}\right]\end{aligned}$
 
-  as per the procedure, comment concluds about consistant/inconsistent from here
 
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<footer style = "font-size: 18px">Example-3 $\rightarrow$ Example-3 $\rightarrow$ <span style = "color:blue"> Example-3 </span> $\rightarrow$ Summary $\rightarrow$ Example-4 </footer>

### <Success style="font-size:25px"> Determinants L-4 </Success>
### <primary style="font-size:24px"> Summary </primary>
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-  Previous example
- 3 variations
- summary,
 
-  1 $\operatorname{det}(A) \neq 0 \Rightarrow \underline{x}=A^{-1} B$ (one point of intersection)

-  2 (b) $\operatorname{det}(A)=0 \rightarrow$ (inconsistent) (parallel lines)

-  2 (a) $\operatorname{det}(A)=0 \rightarrow$ no conclusion ( some line $\Rightarrow$ infinitily many solutions)
 
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<footer style = "font-size: 18px">Example-3 $\rightarrow$ Example-3 $\rightarrow$ <span style = "color:blue"> Summary </span> $\rightarrow$ Example-4 $\rightarrow$ Example-4 </footer>

### <Success style="font-size:25px"> Determinants L-4 </Success>
### <primary style="font-size:24px"> Example-4 </primary>
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- Briefly,

- $ \alpha x+2 y=\lambda$
 
- $  3 x-2 y=\mu$

- $\alpha, \lambda, h \in \mathbb{R}$

-  Is the following true?

-  if $\alpha \neq-3$ then the system has a only one unique solution for all $ \lambda$ & $ \mu $.
 
- $\left[\begin{array}{cc}\alpha & 2 \\\ 3 & -2\end{array}\right]\left[\begin{array}{l}x \\\ y\end{array}\right]=\left[\begin{array}{l}\lambda \\\ \mu\end{array}\right]$
 
- $A \times x = B$
 
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<footer style = "font-size: 18px">Example-3 $\rightarrow$ Summary $\rightarrow$ <span style = "color:blue"> Example-4 </span> $\rightarrow$ Example-4 $\rightarrow$ Thank you </footer>

### <Success style="font-size:25px"> Determinants L-4 </Success>
### <primary style="font-size:24px"> Example-4 </primary>
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- $\operatorname{det}(A)=-2 \alpha-6$

- $\operatorname{det}(A) \neq 0$ if $\alpha \neq-3$ $(\alpha \neq-3)$

-  if $\operatorname{det}(A) \pm 0 \Rightarrow$ consistent system.
 
- $\underline{x}=A^{-1} B=\frac{1}{-2 \alpha-6} \left[\begin{array}{ll}-2 & -2 \\\ -3 & \alpha\end{array}\right]\left[\begin{array}{l}\lambda \\\ \mu\end{array}\right]$
 
- $=\frac{-1}{2 \alpha+6}\left[\begin{array}{l}-2 \lambda-2 \mu \\\ -3 \lambda+\alpha \mu\end{array}\right]$
 
-  one solution for given $\lambda \mu  \alpha(\neq-3)$. Statement is true

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<footer style = "font-size: 18px">Summary $\rightarrow$ Example-4 $\rightarrow$ <span style = "color:blue"> Example-4 </span> $\rightarrow$ Thank you $\rightarrow$  </footer>