mathematics-class-12-unit-04-chapter-09-determinants-l-9-11-oqoofifkpp8

<h3 id="success-stylefont-size25px-determinants-l-9-success-1"><Success style="font-size:25px"> Determinants L-9 </Success></h3>
<h3 id="primary-stylefont-size24px-determinants--area-of-triangles-primary"><primary style="font-size:24px"> Determinants & area of triangles </primary></h3>
<hr>
<main>
<div style='float: left; width:99%;font-size:30px; text-align:left;'>
<ul>
<li>
<p>We know the area of a triangle given by points $\left(x_1, y_1\right)$,$\left(x_2, y_2\right)$ $\left(x_3, y_3\right)$</p>
</li>
<li>
<p>in $\frac{1}{2}\left(x_1\left(y_2-y_3\right)+x_2\left(y_3-y_1\right)\right.$ $+x_3\left(y_1-y_2\right)$ ?</p>
</li>
</ul>
</div>
<div style='float: right; width:1%;'>
		
</div>
</main>
<hr>
<footer style = "font-size: 18px"> $\rightarrow$ Determinants lecture - 9 $\rightarrow$ <span style = "color:blue"> Determinants & area of triangles </span> $\rightarrow$ Example 1 $\rightarrow$ Example 1 </footer>

<h3 id="success-stylefont-size25px-determinants-l-9-success-2"><Success style="font-size:25px"> Determinants L-9 </Success></h3>
<h3 id="primary-stylefont-size24px-example-1-primary"><primary style="font-size:24px"> Example 1 </primary></h3>
<hr>
<main>
<div style='float: left; width:99%;font-size:30px; text-align:left;'>
<ul>
<li>
<p>Consider the following</p>
</li>
<li>
<p>matrix : $ \quad A=\left(\begin{array}{lll}1 & x_1 & y_1 \\ 1 & x_2 & y_2 \\ 1 & x_3 & y_3\end{array}\right)$</p>
</li>
<li>
<p>$\begin{aligned}|A|=1 & \left(x_2 y_3-x_3 y_2\right) -1\left(x_1 y_3-x_3 y_1\right) +1 \left(x_1 y_2-x_2 y_1\right)\end{aligned}$</p>
</li>
</ul>
</div>
<div style='float: right; width:1%;'>
		
</div>
</main>
<hr>
<footer style = "font-size: 18px">Determinants lecture - 9 $\rightarrow$ Determinants & area of triangles $\rightarrow$ <span style = "color:blue"> Example 1 </span> $\rightarrow$ Example 1 $\rightarrow$ Example 1 </footer>

<h3 id="success-stylefont-size25px-determinants-l-9-success-3"><Success style="font-size:25px"> Determinants L-9 </Success></h3>
<h3 id="primary-stylefont-size24px-example-1-primary-1"><primary style="font-size:24px"> Example 1 </primary></h3>
<hr>
<main>
<div style='float: left; width:99%;font-size:30px; text-align:left;'>
<ul>
<li>
<p>$= x_2 y_3-x_3 y_2 - x_1 y_3 + x_3 y_1 + x_1 y_2 - x_2 y_1$</p>
</li>
<li>
<p>$= x_1 (y_2-y_3)+ x_2 (y_3-y_1) + x_3 (y_1-y_3)$</p>
</li>
<li>
<p>$\therefore$ If we compare with the formula for area of a triangle</p>
</li>
</ul>
</div>
<div style='float: right; width:1%;'>
		
</div>
</main>
<hr>
<footer style = "font-size: 18px">Determinants & area of triangles $\rightarrow$ Example 1 $\rightarrow$ <span style = "color:blue"> Example 1 </span> $\rightarrow$ Example 1 $\rightarrow$ Remark </footer>

<h3 id="success-stylefont-size25px-determinants-l-9-success-4"><Success style="font-size:25px"> Determinants L-9 </Success></h3>
<h3 id="primary-stylefont-size24px-example-1-primary-2"><primary style="font-size:24px"> Example 1 </primary></h3>
<hr>
<main>
<div style='float: left; width:99%;font-size:30px; text-align:left;'>
<ul>
<li>
<p>We get</p>
</li>
<li>
<p>$ \Delta = \frac {1}{2}$ $\left|\begin{array}{lll}1 & x_1 & y_1 \\ 1 & x_2 & y_2 \\ 1 & x_3 & y_3\end{array}\right|$</p>
</li>
<li>
<p>$ = \frac {1}{2}$ $\left|\begin{array}{lll}x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{array}\right|$</p>
</li>
<li>
<p>If determinant comes out to be-ve, we take the absolute value for area</p>
</li>
</ul>
</div>
<div style='float: right; width:1%;'>
		
</div>
</main>
<hr>
<footer style = "font-size: 18px">Example 1 $\rightarrow$ Example 1 $\rightarrow$ <span style = "color:blue"> Example 1 </span> $\rightarrow$ Remark $\rightarrow$ Example 2 </footer>

<h3 id="success-stylefont-size25px-determinants-l-9-success-5"><Success style="font-size:25px"> Determinants L-9 </Success></h3>
<h3 id="primary-stylefont-size24px-remark-primary"><primary style="font-size:24px"> Remark </primary></h3>
<hr>
<main>
<div style='float: left; width:99%;font-size:30px; text-align:left;'>
<ul>
<li>
<p>What happens if the determinant in 0</p>
</li>
<li>
<p>Area of a triangle = 0 if the points are collinear</p>
</li>
<li>
<p>$ \therefore $ To test collinearily we may use determinant in the above way</p>
</li>
</ul>
</div>
<div style='float: right; width:1%;'>
		
</div>
</main>
<hr>
<footer style = "font-size: 18px">Example 1 $\rightarrow$ Example 1 $\rightarrow$ <span style = "color:blue"> Remark </span> $\rightarrow$ Example 2 $\rightarrow$ Example 3 </footer>

<h3 id="success-stylefont-size25px-determinants-l-9-success-6"><Success style="font-size:25px"> Determinants L-9 </Success></h3>
<h3 id="primary-stylefont-size24px-example-2-primary"><primary style="font-size:24px"> Example 2 </primary></h3>
<hr>
<main>
<div style='float: left; width:99%;font-size:26px; text-align:left;'>
<ul>
<li>
<p>Show that $\begin{aligned} \quad & A=(a, b+c) \\ & B=(b, c+a) \\ & C=(c, a+b)\end{aligned} \quad $ are Collinear</p>
</li>
<li>
<p>$ \therefore$ We compare with the determinant?</p>
</li>
<li>
<p>$\left|\begin{array}{lll}1 & a & b+c \\ 1 & b & c+a \\ 1 & c & a+b\end{array}\right|$</p>
</li>
<li>
<p>$c_3=c_3+c_2$</p>
</li>
<li>
<p>$= $ $\left|\begin{array}{lll}1 & a & a+b+c \\ 1 & b & a+b+c \\ 1 & c & a+b+c\end{array}\right|$</p>
</li>
<li>
<p>$ = (a+b+c)$ $\left(\begin{array}{lll}1 & a & 1 \\ 1 & b & 1 \\ 1 & c & 1\end{array}\right)$</p>
</li>
<li>
<p>$ = 0 $</p>
</li>
</ul>
</div>
<div style='float: right; width:1%;'>
		
</div>
</main>
<hr>
<footer style = "font-size: 18px">Example 1 $\rightarrow$ Remark $\rightarrow$ <span style = "color:blue"> Example 2 </span> $\rightarrow$ Example 3 $\rightarrow$ Example 3 </footer>

<h3 id="success-stylefont-size25px-determinants-l-9-success-7"><Success style="font-size:25px"> Determinants L-9 </Success></h3>
<h3 id="primary-stylefont-size24px-example-3-primary"><primary style="font-size:24px"> Example 3 </primary></h3>
<hr>
<main>
<div style='float: left; width:99%;font-size:30px; text-align:left;'>
<ul>
<li>
<p>If points</p>
</li>
<li>
<p>(a, 0) (0, b) & (1, 1)</p>
</li>
<li>
<p>are collinear then show that a+b = ab</p>
</li>
<li>
<p>since there are collinear</p>
</li>
<li>
<p>$\left(\begin{array}{lll}1 & a & 0 \\ 1 & 0 & b \\ 1 & 1 & 1\end{array}\right)$ $ = 0$</p>
</li>
</ul>
</div>
<div style='float: right; width:1%;'>
		
</div>
</main>
<hr>
<footer style = "font-size: 18px">Remark $\rightarrow$ Example 2 $\rightarrow$ <span style = "color:blue"> Example 3 </span> $\rightarrow$ Example 3 $\rightarrow$ Example 3 </footer>

<h3 id="success-stylefont-size25px-determinants-l-9-success-8"><Success style="font-size:25px"> Determinants L-9 </Success></h3>
<h3 id="primary-stylefont-size24px-example-3-primary-1"><primary style="font-size:24px"> Example 3 </primary></h3>
<hr>
<main>
<div style='float: left; width:99%;font-size:30px; text-align:left;'>
<ul>
<li>
<p>By $ R_1 \leftarrow R_1 - R_2 $</p>
</li>
<li>
<p>We get the determinant in</p>
</li>
<li>
<p>$\left|\begin{array}{lll}0 & R & -b \\ 1 & 0 & b \\ 1 & 1 & 1\end{array}\right|$</p>
</li>
<li>
<p>Now $ R_3 \leftarrow R_3 - R_2 $</p>
</li>
<li>
<p>We have $\left|\begin{array}{lll}0 & R & -b \\ 1 & 0 & b \\ 0 & 1 & 1-b\end{array}\right|$</p>
</li>
</ul>
</div>
<div style='float: right; width:1%;'>
		
</div>
</main>
<hr>
<footer style = "font-size: 18px">Example 2 $\rightarrow$ Example 3 $\rightarrow$ <span style = "color:blue"> Example 3 </span> $\rightarrow$ Example 3 $\rightarrow$ Example 4 </footer>

<h3 id="success-stylefont-size25px-determinants-l-9-success-10"><Success style="font-size:25px"> Determinants L-9 </Success></h3>
<h3 id="primary-stylefont-size24px-example-4-primary"><primary style="font-size:24px"> Example 4 </primary></h3>
<hr>
<main>
<div style='float: left; width:99%;font-size:30px; text-align:left;'>
<ul>
<li>
<p>Use deterninant method to find equation of the line joining points</p>
</li>
<li>
<p>$A = (-2, 4), B= (2,-6)$</p>
</li>
<li>
<p>We know from our knowladge of coordinate geometry:</p>
</li>
<li>
<p>$\frac{y-4}{4+6}=\frac{x+2}{-2-2}$ or $\frac{y-y}{10}=\frac{x+2}{-4}$</p>
</li>
<li>
<p>or $10x+20 =-4y$+16</p>
</li>
<li>
<p>or $10x+4y+4 = 0$</p>
</li>
</ul>
</div>
<div style='float: right; width:1%;'>
		
</div>
</main>
<hr>
<footer style = "font-size: 18px">Example 3 $\rightarrow$ Example 3 $\rightarrow$ <span style = "color:blue"> Example 4 </span> $\rightarrow$ Example 4 $\rightarrow$ Example 4 </footer>

<h3 id="success-stylefont-size25px-determinants-l-9-success-11"><Success style="font-size:25px"> Determinants L-9 </Success></h3>
<h3 id="primary-stylefont-size24px-example-4-primary-1"><primary style="font-size:24px"> Example 4 </primary></h3>
<hr>
<main>
<div style='float: left; width:99%;font-size:30px; text-align:left;'>
<ul>
<li>
<p>Let x, y be any point on the line joining $(-2, 4) \& (2, -6)$</p>
</li>
<li>
<p>$\therefore$ $\left|\begin{array}{lll}1 & -2 & 4 \\ 1 & 2 & -6 \\ 1 & x & y\end{array}\right| = 0$</p>
</li>
<li>
<p>or $\left|\begin{array}{lll}0 & -4 & 10 \\ 1 & 2 & -6 \\ 1 & x & y\end{array}\right|=0  R_1 \leftarrow R_1 - R_2$</p>
</li>
<li>
<p>or $\left|\begin{array}{lll}0 & -4 & 10 \\ 0 & 2-x & -6-y \\ 1 & x & y\end{array}\right|=0  R_2 \leftarrow R_2 - R_3$</p>
</li>
</ul>
</div>
<div style='float: right; width:1%;'>
		
</div>
</main>
<hr>
<footer style = "font-size: 18px">Example 3 $\rightarrow$ Example 4 $\rightarrow$ <span style = "color:blue"> Example 4 </span> $\rightarrow$ Example 4 $\rightarrow$ Example 5 </footer>

<h3 id="success-stylefont-size25px-determinants-l-9-success-12"><Success style="font-size:25px"> Determinants L-9 </Success></h3>
<h3 id="primary-stylefont-size24px-example-4-primary-2"><primary style="font-size:24px"> Example 4 </primary></h3>
<hr>
<main>
<div style='float: left; width:99%;font-size:30px; text-align:left;'>
<ul>
<li>
<p>or $(-4)(-6-y) - 10(2-x) = 0$</p>
</li>
<li>
<p>or $24 + 4y - 20 + 10x = 0$</p>
</li>
<li>
<p>or $10x + 4y + 4 = 0 $</p>
</li>
<li>
<p>$ (-2,4) \& (2,-6)$</p>
</li>
</ul>
</div>
<div style='float: right; width:1%;'>
		
</div>
</main>
<hr>
<footer style = "font-size: 18px">Example 4 $\rightarrow$ Example 4 $\rightarrow$ <span style = "color:blue"> Example 4 </span> $\rightarrow$ Example 5 $\rightarrow$ Example 5 </footer>

<h3 id="success-stylefont-size25px-determinants-l-9-success-13"><Success style="font-size:25px"> Determinants L-9 </Success></h3>
<h3 id="primary-stylefont-size24px-example-5-primary"><primary style="font-size:24px"> Example 5 </primary></h3>
<hr>
<main>
<div style='float: left; width:99%;font-size:30px; text-align:left;'>
<ul>
<li>
<p>If the area of the triangle $A(-2,4), \quad B(2,-6)$ & $C(5, k)$ in 35 units what is the value & $k$ ?</p>
</li>
<li>
<p>i.e $\frac{1}{2}\left|\begin{array}{ccc}1 & -2 & 4 \\ 1 & 2 & -6 \\ 1 & 5 & k\end{array}\right|=35 \Rightarrow\left|\begin{array}{rrr}1 & -2 & 4 \\ 1 & 2 & -6 \\ 1 & 5 & k\end{array}\right|=70$</p>
</li>
<li>
<p>or $\left|\begin{array}{ccc}0 & -4 & 10 \\ 1 & 2 & -6 \\ 1 & 5 & k\end{array}\right|=\begin{aligned} & 70, \  R_1 \rightarrow R_1-R_2\end{aligned}$</p>
</li>
</ul>
</div>
<div style='float: right; width:1%;'>
	
		
</div>
</main>
<hr>
<footer style = "font-size: 18px">Example 4 $\rightarrow$ Example 4 $\rightarrow$ <span style = "color:blue"> Example 5 </span> $\rightarrow$ Example 5 $\rightarrow$  Minors & Cofactors </footer>

<h3 id="success-stylefont-size25px-determinants-l-9-success-14"><Success style="font-size:25px"> Determinants L-9 </Success></h3>
<h3 id="primary-stylefont-size24px-example-5-primary-1"><primary style="font-size:24px"> Example 5 </primary></h3>
<hr>
<main>
<div style='float: left; width:99%;font-size:30px; text-align:left;'>
<ul>
<li>
<p>or $\left|\begin{array}{ccc}0 & -4 & 10 \\ 1 & 2 & -6 \\ 0 & 3 & k+6\end{array}\right|=\begin{aligned} & 70, \  R_3 \rightarrow R_3-R_2\end{aligned}$</p>
</li>
<li>
<p>or $(-1) (-4(k+6)-3.10) = 70$</p>
</li>
<li>
<p>or $(-1) (-4k-24-30) = 70, or \quad 4k+24+30 = 70, or \quad 4k = 16, or \quad k = 4$</p>
</li>
</ul>
</div>
<div style='float: right; width:1%;'>
		
		
</div>
</main>
<hr>
<footer style = "font-size: 18px">Example 4 $\rightarrow$ Example 5 $\rightarrow$ <span style = "color:blue"> Example 5 </span> $\rightarrow$  Minors & Cofactors $\rightarrow$  Minors & Cofactors </footer>

<h3 id="success-stylefont-size25px-determinants-l-9-success-15"><Success style="font-size:25px"> Determinants L-9 </Success></h3>
<h3 id="primary-stylefont-size24px--minors--cofactors-primary"><primary style="font-size:24px">  Minors & Cofactors </primary></h3>
<hr>
<main>
<div style='float: left; width:99%;font-size:30px; text-align:left;'>
<ul>
<li>
<p>We have expresed the determinant of a matrix A as $\sum_{j=1}^n a_{1 j} \cdot(-1)^{1+i} M_{i j}$</p>
</li>
<li>
<p>Where $M_{1j}$ in the determinant of the submatrix of A after deleting Row 1 & column J</p>
</li>
</ul>
</div>
<div style='float: right; width:1%;'>
		
</div>
</main>
<hr>
<footer style = "font-size: 18px">Example 5 $\rightarrow$ Example 5 $\rightarrow$ <span style = "color:blue">  Minors & Cofactors </span> $\rightarrow$  Minors & Cofactors $\rightarrow$ Minors </footer>

<h3 id="success-stylefont-size25px-determinants-l-9-success-16"><Success style="font-size:25px"> Determinants L-9 </Success></h3>
<h3 id="primary-stylefont-size24px--minors--cofactors-primary-1"><primary style="font-size:24px">  Minors & Cofactors </primary></h3>
<hr>
<main>
<div style='float: left; width:99%;font-size:30px; text-align:left;'>
<ul>
<li>
<p>In a similar way we can also write :</p>
</li>
<li>
<p>$|A| = \sum_j a_{i j}(-1)^{i+j} M_{i j}$</p>
</li>
<li>
<p>by expanding along row j This $M_{ij}$ in called the Minor corresponding the element $a_{ij}$</p>
</li>
</ul>
</div>
<div style='float: right; width:1%;'>
		
</div>
</main>
<hr>
<footer style = "font-size: 18px">Example 5 $\rightarrow$  Minors & Cofactors $\rightarrow$ <span style = "color:blue">  Minors & Cofactors </span> $\rightarrow$ Minors $\rightarrow$ Cofactors </footer>

<h3 id="success-stylefont-size25px-determinants-l-9-success-17"><Success style="font-size:25px"> Determinants L-9 </Success></h3>
<h3 id="primary-stylefont-size24px-minors-primary"><primary style="font-size:24px"> Minors </primary></h3>
<hr>
<main>
<div style='float: left; width:99%;font-size:30px; text-align:left;'>
<ul>
<li>Def: For a given matrix $A$ if we consider the element</li>
<li>$a_{ij}$ (i.e at the position $i^{th}$ row & $j^{th}$ column then corresponding minor $M_{ij}$ is the determinant of the $n-1 \times n-1$ matrix, obtained after deleting $i^{th}$ row & $j^{th}$ column of $A$</li>
</ul>
</div>
<div style='float: right; width:1%;'>
		
</div>
</main>
<hr>
<footer style = "font-size: 18px"> Minors & Cofactors $\rightarrow$  Minors & Cofactors $\rightarrow$ <span style = "color:blue"> Minors </span> $\rightarrow$ Cofactors $\rightarrow$ Example 6 </footer>

<h3 id="success-stylefont-size25px-determinants-l-9-success-20"><Success style="font-size:25px"> Determinants L-9 </Success></h3>
<h3 id="primary-stylefont-size24px-example-7-primary"><primary style="font-size:24px"> Example 7 </primary></h3>
<hr>
<main>
<div style='float: left; width:99%;font-size:30px; text-align:left;'>
<ul>
<li>
<p>Example If A = $\left(\begin{array}{lll}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}\end{array}\right)$</p>
</li>
<li>
<p>which of the following given |A|?</p>
</li>
<li>
<p>(A) $a_{21} M_{21}+a_{22} M_{22}+a_{23} M_{23}$</p>
</li>
<li>
<p>(B) $a_{11} M_{11}-a_{12} M_{21}+a_{13} M_{31}$</p>
</li>
<li>
<p>(C) $a_{31} A_{31}-a_{32} A_{32}+a_{33} A_{33}$</p>
</li>
<li>
<p>(D) $a_{13} A_{13}+a_{23} A_{23}+a_{33} A_{33}$</p>
</li>
<li>
<p>Answer = (D)</p>
</li>
</ul>
</div>
<div style='float: right; width:1%;'>
		
</div>
</main>
<hr>
<footer style = "font-size: 18px">Cofactors $\rightarrow$ Example 6 $\rightarrow$ <span style = "color:blue"> Example 7 </span> $\rightarrow$ Adjoint of a Matrix $\rightarrow$ Example 8 </footer>

<h3 id="success-stylefont-size25px-determinants-l-9-success-21"><Success style="font-size:25px"> Determinants L-9 </Success></h3>
<h3 id="primary-stylefont-size24px-adjoint-of-a-matrix-primary"><primary style="font-size:24px"> Adjoint of a Matrix </primary></h3>
<hr>
<main>
<div style='float: left; width:99%;font-size:30px; text-align:left;'>
<ul>
<li>
<p>Def: The adjoint of a square matrix $A$ is defined as the transpose of (( $\left.A_{i j}\right)$ ) where $A_{i j}$ in the cofacter of $G_{i j}$.</p>
</li>
<li>
<p>i.e $A=\left(\begin{array}{lll}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}\end{array}\right)$</p>
</li>
<li>
<p>Then adj (A) is the 3 $\times$ 3 Matrix :</p>
</li>
<li>
<p>$\left[\begin{array}{lll}A_{11} & A_{21} & A_{31} \\ A_{12} & A_{22} & A_{32} \\ A_{13} & A_{23} & A_{33}\end{array}\right]$</p>
</li>
</ul>
</div>
<div style='float: right; width:1%;'>
		
</div>
</main>
<hr>
<footer style = "font-size: 18px">Example 6 $\rightarrow$ Example 7 $\rightarrow$ <span style = "color:blue"> Adjoint of a Matrix </span> $\rightarrow$ Example 8 $\rightarrow$ Example 8 </footer>

<h3 id="success-stylefont-size25px-determinants-l-9-success-22"><Success style="font-size:25px"> Determinants L-9 </Success></h3>
<h3 id="primary-stylefont-size24px-example-8-primary"><primary style="font-size:24px"> Example 8 </primary></h3>
<hr>
<main>
<div style='float: left; width:99%;font-size:30px; text-align:left;'>
<ul>
<li>
<p>If A = $\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]$</p>
</li>
<li>
<p>Then</p>
</li>
<li>
<p>$A_{11} =  d$ $\quad$ $A_{12} = -c$</p>
</li>
<li>
<p>$A_{21} = -b$ $\quad$ $A_{22} =  a$</p>
</li>
</ul>
</div>
<div style='float: right; width:1%;'>
	
</div>
</main>
<hr>
<footer style = "font-size: 18px">Example 7 $\rightarrow$ Adjoint of a Matrix $\rightarrow$ <span style = "color:blue"> Example 8 </span> $\rightarrow$ Example 8 $\rightarrow$ Example 9 </footer>

<h3 id="success-stylefont-size25px-determinants-l-9-success-23"><Success style="font-size:25px"> Determinants L-9 </Success></h3>
<h3 id="primary-stylefont-size24px-example-8-primary-1"><primary style="font-size:24px"> Example 8 </primary></h3>
<hr>
<main>
<div style='float: left; width:99%;font-size:30px; text-align:left;'>
<ul>
<li>
<p>$\therefore$ $\operatorname{adj}(A)=\left[\begin{array}{cc}d & -b \\ -c & a\end{array}\right]$</p>
</li>
<li>
<p>i.e the diagonal element of A are interchange & 2 sign off- off-dingonal elements are changes</p>
</li>
<li>
<p>Thus for a $2 \times 2$ matrix we get the adjoint very easily</p>
</li>
</ul>
</div>
<div style='float: right; width:1%;'>
		
</div>
</main>
<hr>
<footer style = "font-size: 18px">Adjoint of a Matrix $\rightarrow$ Example 8 $\rightarrow$ <span style = "color:blue"> Example 8 </span> $\rightarrow$ Example 9 $\rightarrow$ Example 9 </footer>

<h3 id="success-stylefont-size25px-determinants-l-9-success-24"><Success style="font-size:25px"> Determinants L-9 </Success></h3>
<h3 id="primary-stylefont-size24px-example-9-primary"><primary style="font-size:24px"> Example 9 </primary></h3>
<hr>
<main>
<div style='float: left; width:99%;font-size:30px; text-align:left;'>
<ul>
<li>
<p>What is A .(adj (A))?</p>
</li>
<li>
<p>$\left[\begin{array}{ll}a & b \\ c & d\end{array}\right] \cdot\left[\begin{array}{cc}d & -b \\ -c & a\end{array}\right]$</p>
</li>
<li>
<p>$=\left[\begin{array}{cc}a d-b c & -a b+a b \\ c d-c d & a d-b c\end{array}\right]$</p>
</li>
<li>
<p>$=\left[\begin{array}{cc}|A| & 0 \\ 0 & |A|\end{array}\right]$</p>
</li>
<li>
<p>i.e it is a diagonal matrix with diagonal entries $=|A|$</p>
</li>
</ul>
</div>
<div style='float: right; width:1%;'>
		
</div>
</main>
<hr>
<footer style = "font-size: 18px">Example 8 $\rightarrow$ Example 8 $\rightarrow$ <span style = "color:blue"> Example 9 </span> $\rightarrow$ Example 9 $\rightarrow$ Example 10 </footer>

<h3 id="success-stylefont-size25px-determinants-l-9-success-26"><Success style="font-size:25px"> Determinants L-9 </Success></h3>
<h3 id="primary-stylefont-size24px-example-10-primary"><primary style="font-size:24px"> Example 10 </primary></h3>
<hr>
<main>
<div style='float: left; width:99%;font-size:30px; text-align:left;'>
<ul>
<li>
<p>Let us verify the result for $3 \times 3$ matrix</p>
</li>
<li>
<p>Let $P$ be the product of P = A (adj(A))</p>
</li>
<li>
<p>i. $e \quad P=\left(\begin{array}{lll}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}\end{array}\right)\left(\begin{array}{lll}A_{11} & A_{21} & A_{31} \\ A_{12} & A_{22} & A_{32} \\ A_{13} & A_{23} & A_{33}\end{array}\right)$</p>
</li>
</ul>
</div>
<div style='float: right; width:1%;'>
		
</div>
</main>
<hr>
<footer style = "font-size: 18px">Example 9 $\rightarrow$ Example 9 $\rightarrow$ <span style = "color:blue"> Example 10 </span> $\rightarrow$ Example 10 $\rightarrow$ Example 10 </footer>

<h3 id="success-stylefont-size25px-determinants-l-9-success-27"><Success style="font-size:25px"> Determinants L-9 </Success></h3>
<h3 id="primary-stylefont-size24px-example-10-primary-1"><primary style="font-size:24px"> Example 10 </primary></h3>
<hr>
<main>
<div style='float: left; width:99%;font-size:30px; text-align:left;'>
<ul>
<li>
<p>We want to verify the</p>
</li>
<li>
<p>$P=|A|\left(\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right)$</p>
</li>
<li>
<p>Let us look at the</p>
</li>
<li>
<p>$\begin{aligned} P_{11} & =a_{11} A_{11}+a_{12} A_{12}+a_{13} A_{13} \ & =|A|\end{aligned}$</p>
</li>
<li>
<p>Similarly $P_{22}$ & $P_{33}$ can be shown to be |A|</p>
</li>
</ul>
</div>
<div style='float: right; width:1%;'>
	
</div>
</main>
<hr>
<footer style = "font-size: 18px">Example 9 $\rightarrow$ Example 10 $\rightarrow$ <span style = "color:blue"> Example 10 </span> $\rightarrow$ Example 10 $\rightarrow$ Example 10 </footer>

<h3 id="success-stylefont-size25px-determinants-l-9-success-28"><Success style="font-size:25px"> Determinants L-9 </Success></h3>
<h3 id="primary-stylefont-size24px-example-10-primary-2"><primary style="font-size:24px"> Example 10 </primary></h3>
<hr>
<main>
<div style='float: left; width:99%;font-size:30px; text-align:left;'>
<ul>
<li>
<p>We need to show that off diagonal elements are 0 . $p_{12}=a_{11} A_{21}+a_{12} A_{22}+a_{13} A_{23}$</p>
</li>
<li>
<p>$=a_{11}(-1)^{2+1}\left|\begin{array}{ll}a_{12} & a_{13} \\ a_{32} & a_{33}\end{array}\right|$</p>
</li>
<li>
<p>$+a_{12}(-1)^{2+2}\left|\begin{array}{ll}a_{11} & a_{13} \\ a_{31} & a_{33}\end{array}\right|$</p>
</li>
<li>
<p>$+a_{13}(-1)^{2+3}\left|\begin{array}{ll}a_{11} & a_{12} \\ a_{31} & a_{32}\end{array}\right|$</p>
</li>
</ul>
</div>
<div style='float: right; width:1%;'>
	
</div>
</main>
<hr>
<footer style = "font-size: 18px">Example 10 $\rightarrow$ Example 10 $\rightarrow$ <span style = "color:blue"> Example 10 </span> $\rightarrow$ Example 10 $\rightarrow$ Thank you </footer>

<h3 id="success-stylefont-size25px-determinants-l-9-success-29"><Success style="font-size:25px"> Determinants L-9 </Success></h3>
<h3 id="primary-stylefont-size24px-example-10-primary-3"><primary style="font-size:24px"> Example 10 </primary></h3>
<hr>
<main>
<div style='float: left; width:99%;font-size:30px; text-align:left;'>
<ul>
<li>
<p>$= -a_{11} a_{12} a_{33}+a_{11} a_{13} a_{32}$ $+{a_{12} a_{11} a_{33}}{-a_{12} a_{13} a_{31} }$ $\begin{aligned} & -a_{13} a_{11} a_{32} \ +a_{13} a_{31} a_{12} \end{aligned}$ = 0</p>
</li>
<li>
<p>Verfied for $P_{12}$</p>
</li>
<li>
<p>$ \therefore$ A (adj A) = |A| $I_{3}$</p>
</li>
</ul>
</div>
<div style='float: right; width:1%;'>
		
</div>
</main>
<hr>
<footer style = "font-size: 18px">Example 10 $\rightarrow$ Example 10 $\rightarrow$ <span style = "color:blue"> Example 10 </span> $\rightarrow$ Thank you $\rightarrow$  </footer>