mathematics-class-12-unit-04-chapter-01-determinants-l-1-11-sgtjrcglc-g

<h3 id="success-stylefont-size25px-determinants-l-1-success-1"><Success style="font-size:25px"> Determinants L-1 </Success></h3>
<h3 id="primary-stylefont-size24px-determinants-primary"><primary style="font-size:24px"> Determinants </primary></h3>
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<ul>
<li>
<p>a ‘useful’ number associated with a square matrix</p>
</li>
<li>
<p>Example: System of equations-</p>
</li>
<li>
<p>$
\begin{gathered}
x+y=10  \\
4 x=y
\end{gathered}$</p>
</li>
</ul>
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<footer style = "font-size: 18px"> $\rightarrow$ Determinants lecture 1 $\rightarrow$ <span style = "color:blue"> Determinants </span> $\rightarrow$ Determinants $2 \times 2$ $\rightarrow$ Determinants $2 \times 2$ </footer>

<h3 id="success-stylefont-size25px-determinants-l-1-success-2"><Success style="font-size:25px"> Determinants L-1 </Success></h3>
<h3 id="primary-stylefont-size24px-determinants-2-times-2-primary"><primary style="font-size:24px"> Determinants $2 \times 2$ </primary></h3>
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<div style='float: left; width:99%;font-size:30px; text-align:left;'>
<ul>
<li>
<p>$\Rightarrow  x+y=10 $</p>
</li>
<li>
<p>$\Rightarrow 4 x-y=0$</p>
</li>
<li>
<p>$\left[\begin{array}{ll}
1 & 1 \\
4 & -1
\end{array}\right]\left[\begin{array}{l}
x \\
y
\end{array}\right]$= $\left[\begin{array}{c}
10 \\
0
\end{array}\right]$</p>
</li>
<li>
<p>$\left[\begin{array}{ll}
a & b \\
c & d
\end{array}\right]\left[\begin{array}{l}
x \\
y
\end{array}\right]$= $\left[\begin{array}{c}
M \\
N
\end{array}\right]$</p>
</li>
<li>
<p>$\Rightarrow 2 \times 2$ square matrix</p>
</li>
</ul>
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<hr>
<footer style = "font-size: 18px">Determinants lecture 1 $\rightarrow$ Determinants $\rightarrow$ <span style = "color:blue"> Determinants $2 \times 2$ </span> $\rightarrow$ Determinants $2 \times 2$ $\rightarrow$ Determinants $2 \times 2$ </footer>

<h3 id="success-stylefont-size25px-determinants-l-1-success-3"><Success style="font-size:25px"> Determinants L-1 </Success></h3>
<h3 id="primary-stylefont-size24px-determinants-2-times-2-primary-1"><primary style="font-size:24px"> Determinants $2 \times 2$ </primary></h3>
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<div style='float: left; width:99%;font-size:30px; text-align:left;'>
<ul>
<li>
<p>$
\begin{gathered}
d \times[a x+b y=M \\ -b \times[c x+d y=N \\
\Rightarrow(a d-b c) x=d M-b N \\
\text { if } a d-b c \neq 0 \\
\Rightarrow x=\frac{d M-b N}{a d-b c}
\end{gathered}
$</p>
</li>
<li>
<p>Similarly , for $y$,</p>
</li>
<li>
<p>$
\begin{aligned}
a x+b y & =M \\
c x+d y & =N
\end{aligned}
$</p>
</li>
</ul>
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<footer style = "font-size: 18px">Determinants $\rightarrow$ Determinants $2 \times 2$ $\rightarrow$ <span style = "color:blue"> Determinants $2 \times 2$ </span> $\rightarrow$ Determinants $2 \times 2$ $\rightarrow$ Determinants as an area </footer>

### <Success style="font-size:25px"> Determinants L-1 </Success>
### <primary style="font-size:24px"> Determinants $2 \times 2$ </primary>
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- $ \Rightarrow y=\frac{c M-a N}{b c-a d} $

- This is for $a d-b c \neq 0$

- $ 
\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]
 $

- solutions found when $a d-b c \neq 0$

- Thus, this is the determinant of the $2\times2$ matrix $\left[\begin{array}{ll}a & b  \\\\ c & d\end{array}\right]$

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<hr>
<footer style = "font-size: 18px">Determinants $2 \times 2$ $\rightarrow$ Determinants $2 \times 2$ $\rightarrow$ <span style = "color:blue"> Determinants $2 \times 2$ </span> $\rightarrow$ Determinants as an area $\rightarrow$ Determinants as an area </footer>

<h3 id="success-stylefont-size25px-determinants-l-1-success-4"><Success style="font-size:25px"> Determinants L-1 </Success></h3>
<h3 id="primary-stylefont-size24px-determinants-as-an-area-primary"><primary style="font-size:24px"> Determinants as an area </primary></h3>
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<div style='float: left; width:50%;font-size:30px; text-align:left;'>
<ul>
<li>
<p>Determinant provides a way to check for the existence of solution in a linear system of equations</p>
</li>
<li>
<p>Example 2:  deteteminant as an area
$
\begin{aligned}
& A=\left[\begin{array}{l}
a \\
c
\end{array}\begin{array}{l}
b \\
d
\end{array}\right] \\
& \text { area }=a d-b c \
&
\end{aligned}
$</p>
</li>
</ul>
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</div>
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<hr>
<footer style = "font-size: 18px">Determinants $2 \times 2$ $\rightarrow$ Determinants $2 \times 2$ $\rightarrow$ <span style = "color:blue"> Determinants as an area </span> $\rightarrow$ Determinants as an area $\rightarrow$ Determinants as an area </footer>

<h3 id="success-stylefont-size25px-determinants-l-1-success-5"><Success style="font-size:25px"> Determinants L-1 </Success></h3>
<h3 id="primary-stylefont-size24px-determinants-as-an-area-primary-1"><primary style="font-size:24px"> Determinants as an area </primary></h3>
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<div style='float: left; width:50%;font-size:30px; text-align:left;'>
<ul>
<li>$
\begin{aligned}
&\begin{aligned}
\Delta= & (a+b)(c+d) \\
& -\frac{1}{2} a c-b c\\-\frac{1}{2} b d
& -\frac{1}{2} b d-b c\\-\frac{1}{2} a c
\end{aligned}\
\end{aligned}
$</li>
</ul>
</div>
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<hr>
<footer style = "font-size: 18px">Determinants $2 \times 2$ $\rightarrow$ Determinants as an area $\rightarrow$ <span style = "color:blue"> Determinants as an area </span> $\rightarrow$ Determinants as an area $\rightarrow$ Determinants example </footer>

<h3 id="success-stylefont-size25px-determinants-l-1-success-6"><Success style="font-size:25px"> Determinants L-1 </Success></h3>
<h3 id="primary-stylefont-size24px-determinants-as-an-area-primary-2"><primary style="font-size:24px"> Determinants as an area </primary></h3>
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<div style='float: left; width:50%;font-size:30px; text-align:left;'>
<ul>
<li>
<p>$
\begin{aligned}
& \Delta=(a+b)(c+d)-a c-b d-2 b c \\
&=a c+a d+b c+b d-a c-b d-2 b c \\
&=a d-b c \\
& {\left[\begin{array}{ll}
a & b \\
c & d
\end{array}\right] \rightarrow \Delta = a d-b c }
\end{aligned}
$</p>
</li>
<li>
<p>Determinant gives the ‘area’ of parallelogram formed from the columns of a matrix</p>
</li>
</ul>
</div>
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<hr>
<footer style = "font-size: 18px">Determinants as an area $\rightarrow$ Determinants as an area $\rightarrow$ <span style = "color:blue"> Determinants as an area </span> $\rightarrow$ Determinants example $\rightarrow$ Determinants </footer>

<h3 id="success-stylefont-size25px-determinants-l-1-success-7"><Success style="font-size:25px"> Determinants L-1 </Success></h3>
<h3 id="primary-stylefont-size24px-determinants-example-primary"><primary style="font-size:24px"> Determinants example </primary></h3>
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<div style='float: left; width:99%;font-size:30px; text-align:left;'>
<ul>
<li>
<p>Go back to initial example</p>
</li>
<li>
<p>$
\left[\begin{array}{ll}
1 &  1 \\
4 & -1
\end{array}\right]_{2\times2}\left[\begin{array}{l}
x \\
y
\end{array}\right]=\left[\begin{array}{l}
10  \\
0
\end{array}\right]
$</p>
</li>
<li>
<p>determinant is</p>
</li>
<li>
<p>$1 \times(-1)-1 \times 4$</p>
</li>
<li>
<p>$=-1-4=-5 \neq 0$</p>
</li>
<li>
<p>$\therefore$ Solution exists.</p>
</li>
<li>
<p>Also, area is $|-5|$(absolute value)$=5$</p>
</li>
</ul>
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<hr>
<footer style = "font-size: 18px">Determinants as an area $\rightarrow$ Determinants as an area $\rightarrow$ <span style = "color:blue"> Determinants example </span> $\rightarrow$ Determinants $\rightarrow$ Determinants </footer>

<h3 id="success-stylefont-size25px-determinants-l-1-success-8"><Success style="font-size:25px"> Determinants L-1 </Success></h3>
<h3 id="primary-stylefont-size24px-determinants-primary-1"><primary style="font-size:24px"> Determinants </primary></h3>
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<div style='float: left; width:99%;font-size:30px; text-align:left;'>
<ul>
<li>
<p>Notation: $\operatorname{det}(A)$ or $|A|$</p>
</li>
<li>
<p>$A$ is a square matrix</p>
</li>
<li>
<p>$\longrightarrow$ Define a determinant</p>
</li>
<li>
<p>$\longrightarrow$ Properties</p>
</li>
<li>
<p>$\longrightarrow$ Applications</p>
</li>
</ul>
</div>
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<hr>
<footer style = "font-size: 18px">Determinants as an area $\rightarrow$ Determinants example $\rightarrow$ <span style = "color:blue"> Determinants </span> $\rightarrow$ Determinants $\rightarrow$ Example </footer>

<h3 id="success-stylefont-size25px-determinants-l-1-success-9"><Success style="font-size:25px"> Determinants L-1 </Success></h3>
<h3 id="primary-stylefont-size24px-determinants-primary-2"><primary style="font-size:24px"> Determinants </primary></h3>
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<div style='float: left; width:99%;font-size:25px; text-align:left;'>
<ul>
<li>
<p>Definition</p>
</li>
<li>
<p>$A_{n \times n}=\left[a_{i j}\right]_{n\times n}$</p>
</li>
<li>
<p>$i^{th}$row , $j^{th}$ column</p>
</li>
<li>
<p>$
\begin{gathered}
=\left[\begin{array}{cccc}
a_{11} & a_{12} & \cdots & a_{1 n} \\
a_{21} & a_{22} & \cdots & a_{2 n} \\
\vdots & \vdots &  \ddots & \vdots \\
a_{n 1} & a_{n 2} & \cdots & a_{n n}
\end{array}\right]
\end{gathered}
$</p>
</li>
<li>
<p>$\quad \quad \quad j^{th} column$</p>
</li>
<li>
<p>$= i^{th} row\left[\begin{array}{ccc}& \vdots &   \\ \cdots & a_{ij} & \cdots \\ & \vdots &   \end{array}\right]$</p>
</li>
<li>
<p>Goal: $\operatorname{det}(A)$ in terms of $a_{i j}$</p>
  </div>
</li>
</ul>
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<footer style = "font-size: 18px">Determinants example $\rightarrow$ Determinants $\rightarrow$ <span style = "color:blue"> Determinants </span> $\rightarrow$ Example $\rightarrow$ Example </footer>

### <Success style="font-size:25px"> Determinants L-1 </Success>
### <primary style="font-size:24px"> Example </primary>
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<div style='float: left; width:99%;font-size:30px; text-align:left;'>
	
- **Minor**, $M_{i j}$ is associated with each entry $a_{ij}$. It is the determinant of a matrix obtained from $A$ after deleting the $i^{th}$ row and $j^{th}$ column
$ 
\text { Example: } \quad A=\left[\begin{array}{cc}
 a_{11}=1 & a_{12}=1 \\\\
 a_{21}=4 & a_{22}=-1
\end{array}\right]
$
- what is minor of $a_{12}$ i.e. $M_{12}$
$
 M_{12}=\operatorname{det}([4])=4
$
- as we take detaminant of scalar as itself

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<hr>
<footer style = "font-size: 18px">Determinants $\rightarrow$ Determinants $\rightarrow$ <span style = "color:blue"> Example </span> $\rightarrow$ Example $\rightarrow$ Properties </footer>

<h3 id="success-stylefont-size25px-determinants-l-1-success-10"><Success style="font-size:25px"> Determinants L-1 </Success></h3>
<h3 id="primary-stylefont-size24px-example-primary"><primary style="font-size:24px"> Example </primary></h3>
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<div style='float: left; width:99%;font-size:30px; text-align:left;'>
<ul>
<li>
<p><strong>Cofactor</strong> , $A_{ij}$ is defined as
$
A_{i j}=(-1)^{i+j} M_{i j}
$</p>
</li>
<li>
<p>In example before,
$
\begin{aligned}
A_{12} & =(-1)^{1+2} \cdot M_{12} \\
& =(-1)^3 M_{12} \\
& =-M_{12} \\
& =-4 .
\end{aligned}
$</p>
</li>
</ul>
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<hr>
<footer style = "font-size: 18px">Determinants $\rightarrow$ Example $\rightarrow$ <span style = "color:blue"> Example </span> $\rightarrow$ Properties $\rightarrow$ Example </footer>

### <Success style="font-size:25px"> Determinants L-1 </Success>
### <primary style="font-size:24px"> Example </primary>
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<main>

- Example: $1 \times 1$  matrix which is a scalar. Its determinant was taken as the scalar itself

- Example: $2 \times 2$ matrix
$
\left[\begin{array}{ll}
 a & b \\\\
 c & d
\end{array}\right]=a \cdot d+b\cdot (-c)
$
- cofactor of ' $a$ ': $d(-1)^{1+1}=d$

- cofactor of ' $b$ ': $\quad c(-1)^{1+2}=-c$

- $\operatorname{det}(\left[\begin{array}{ll}
 a & b \\\\
 c & d
\end{array}\right])=a d-b c$

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<hr>
<footer style = "font-size: 18px">Example $\rightarrow$ Properties $\rightarrow$ <span style = "color:blue"> Example </span> $\rightarrow$ Example $\rightarrow$ Example </footer>

### <Success style="font-size:25px"> Determinants L-1 </Success>
### <primary style="font-size:24px"> Example </primary>
<hr>
<main>

- Let's expand a column,
$
\begin{aligned}
 {\left[\begin{array}{ll}
 a & b \downarrow \\ 
 c & d 
\end{array}\right] } & =b(-c)+d(a) \\
& =a d-b c ! !
\end{aligned}
$
- which is same as before

- $\longrightarrow$ formal definition of determinant

- $\longrightarrow$ a number to check existance of solutions

- $\longrightarrow$ as area of parallelogram

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<hr>
<footer style = "font-size: 18px">Properties $\rightarrow$ Example $\rightarrow$ <span style = "color:blue"> Example </span> $\rightarrow$ Example $\rightarrow$ Remarks </footer>

<h3 id="success-stylefont-size25px-determinants-l-1-success-11"><Success style="font-size:25px"> Determinants L-1 </Success></h3>
<h3 id="primary-stylefont-size24px-example-primary-1"><primary style="font-size:24px"> Example </primary></h3>
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<div style='float: left; width:99%;font-size:30px; text-align:left;'>
<ul>
<li>
<p>Example: $3 \times 3$ matrix</p>
</li>
<li>
<p>${\left[\begin{array}{ccc}
1 & 0 & 2 \\
3 & -1 & 2 \\
5 & 2 & 0
\end{array}\right]  }$  $\operatorname{det} ?$</p>
</li>
<li>
<p>$\operatorname{det}$ $= 1 \times(-1)^{1+1}\left|\begin{array}{cc}
-1 & 2 \\
2 & 0 \end{array}\right|$ $+0 \times 0
+2 \times(-1)^{1+3}\left|\begin{array}{cc}
3 & -1 \\
5 & 2\end{array}\right|$</p>
</li>
<li>
<p>$= 1(-1 \times 0-2 \times 2)+2(3 \times 2-(-1) \times 5) \\
= -4+2(6+5)\\=-4+22=18 $</p>
</li>
</ul>
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<footer style = "font-size: 18px">Example $\rightarrow$ Example $\rightarrow$ <span style = "color:blue"> Example </span> $\rightarrow$ Remarks $\rightarrow$ Thank you </footer>