mathematics-class-12-unit-04-chapter-06-determinants-l-6-11-5481icm3rpc

<h3 id="success-stylefont-size25px-determinants-l-6-success-6"><Success style="font-size:25px"> Determinants L-6 </Success></h3>
<h3 id="primary-stylefont-size24px-determinants-of-along-the--3rd--column-primary"><primary style="font-size:24px"> Determinants of along the $ 3^{rd} $ column </primary></h3>
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<ul>
<li>
<p>Let me now expand along the $ 3^{rd} $ column</p>
</li>
<li>
<p>$
\begin{aligned}
& A=\left(\begin{array}{lll}1 & 2 & 3 \\ 4 & 5 & 6 \\ 3 & 1 & 2\end{array}\right) \end{aligned}
$</p>
</li>
<li>
<p>$\begin{aligned} & |A|=(-1)^{1+3} 3(4.1-3.5) \\ & +(-1)^{2+3}.6 \cdot(1.1-3.2) \\ & \quad+(-1)^{3+3} 2(1.5-2.4) \\ & =3(4-15)-6(1-6)+2(5-8) \\ & =-33+30-6=-9\end{aligned}$</p>
</li>
</ul>
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<footer style = "font-size: 18px">Example $\rightarrow$ Rule $\rightarrow$ <span style = "color:blue"> Determinants of along the $ 3^{rd} $ column </span> $\rightarrow$ Determinants  along row 1 $\rightarrow$ Determinants  along row 3 </footer>

<h3 id="success-stylefont-size25px-determinants-l-6-success-7"><Success style="font-size:25px"> Determinants L-6 </Success></h3>
<h3 id="primary-stylefont-size24px-determinants--along-row-1-primary"><primary style="font-size:24px"> Determinants  along row 1 </primary></h3>
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<ul>
<li>
<p>Consider a $3 \times 3$ matrix</p>
</li>
<li>
<p>$ 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)$</p>
</li>
<li>
<p>Its determinant when expanded along row 1 is:</p>
</li>
<li>
<p>$
\begin{aligned} & a_{11}(a_{22} a_{33}-a_{23} a_{32}) \\ &-a_{12}(a_{21} a_{33}-a_{23} a_{31}) \\ &+a_{13}(a_{21} a_{32}-a_{22} a_{31})
\end{aligned}
$</p>
</li>
<li>
<p>$ =a_{11} a_{22} a_{33}-a_{11} a_{23} a_{32} $</p>
</li>
<li>
<p>$ -a_{12} a_{21} a_{33}+a_{12} a_{23} a_{31} $</p>
</li>
<li>
<p>$ +a_{13} a_{21} a_{32}-a_{13} a_{22} a_{31} $</p>
</li>
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<footer style = "font-size: 18px">Rule $\rightarrow$ Determinants of along the $ 3^{rd} $ column $\rightarrow$ <span style = "color:blue"> Determinants  along row 1 </span> $\rightarrow$ Determinants  along row 3 $\rightarrow$ Determinants  along the $2^{nd}$ Column </footer>

<h3 id="success-stylefont-size25px-determinants-l-6-success-8"><Success style="font-size:25px"> Determinants L-6 </Success></h3>
<h3 id="primary-stylefont-size24px-determinants--along-row-3-primary"><primary style="font-size:24px"> Determinants  along row 3 </primary></h3>
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<ul>
<li>
<p>Expand along row 3 we have</p>
</li>
<li>
<p>$
\begin{aligned}&|A|=a_{31}(a_{12} a_{23}-a_{13} a_{22}) -a_{32}(a_{11} a_{23}-a_{13} a_{21}) +a_{33}(a_{11} a_{22}-a_{12} a_{21}) \\ &= a_{12} a_{23} a_{31}-a_{13} a_{22} a_{31} +a_{11} a_{23} a_{32} +a_{13} a_{31} a_{32} +a_{11} a_{22} a_{33} - a_{12} a_{21} a_{33}\end{aligned}
$</p>
</li>
</ul>
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<footer style = "font-size: 18px">Determinants of along the $ 3^{rd} $ column $\rightarrow$ Determinants  along row 1 $\rightarrow$ <span style = "color:blue"> Determinants  along row 3 </span> $\rightarrow$ Determinants  along the $2^{nd}$ Column $\rightarrow$ Determinants  along the $2^{nd}$ Column </footer>

<h3 id="success-stylefont-size25px-determinants-l-6-success-9"><Success style="font-size:25px"> Determinants L-6 </Success></h3>
<h3 id="primary-stylefont-size24px-determinants--along-the-2nd-column-primary"><primary style="font-size:24px"> Determinants  along the $2^{nd}$ Column </primary></h3>
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<div style='float: left; width:99%;font-size:30px; text-align:left;'>
<ul>
<li>
<p>Expansion along the $2^{nd}$ Column: $\left(\begin{array}{lll}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{array}\right)$</p>
</li>
<li>
<p>$|A|$ : Expanding along $2^{nd}$ column. $
\begin{aligned}&\quad -a_{12}(a_{21} a_{33}-a_{23} a_{31})  \quad +a_{22}(a_{11} a_{33}-a_{13}a_{31}) \quad -a_{32}(a_{11} a_{23}-a_{13} a_{21}) \\ &=-a_{12} a_{21} a_{33}+a_{12} a_{23} a_{31} \quad +a_{11} a_{22} a_{33}-a_{13} a_{22} a_{31} \quad -a_{11} a_{23} a_{32}+a_{13} a_{21} a_{32} \end{aligned}
$</p>
</li>
</ul>
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<footer style = "font-size: 18px">Determinants  along row 1 $\rightarrow$ Determinants  along row 3 $\rightarrow$ <span style = "color:blue"> Determinants  along the $2^{nd}$ Column </span> $\rightarrow$ Determinants  along the $2^{nd}$ Column $\rightarrow$ Properties of Determinant-1 </footer>

<h3 id="success-stylefont-size25px-determinants-l-6-success-10"><Success style="font-size:25px"> Determinants L-6 </Success></h3>
<h3 id="primary-stylefont-size24px-determinants--along-the-2nd-column-primary-1"><primary style="font-size:24px"> Determinants  along the $2^{nd}$ Column </primary></h3>
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<ul>
<li>$
\begin{aligned}&\quad -a_{12}(a_{21} a_{33}-a_{23} a_{31}) \quad+a_{22}(a_{11} a_{33}-a_{13}a_{31}) \quad -a_{32}(a_{11} a_{23}-a_{13} a_{21}) \\ &=-a_{12} a_{21} a_{33}+a_{12} a_{23} a_{31} \quad +a_{11} a_{22} a_{33}-a_{13} a_{22} a_{31}  \quad -a_{11} a_{23} a_{32}+a_{13} a_{21} a_{32} \end{aligned}
$</li>
</ul>
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<footer style = "font-size: 18px">Determinants  along row 3 $\rightarrow$ Determinants  along the $2^{nd}$ Column $\rightarrow$ <span style = "color:blue"> Determinants  along the $2^{nd}$ Column </span> $\rightarrow$ Properties of Determinant-1 $\rightarrow$ Example </footer>

<h3 id="success-stylefont-size25px-determinants-l-6-success-12"><Success style="font-size:25px"> Determinants L-6 </Success></h3>
<h3 id="primary-stylefont-size24px-example-primary-2"><primary style="font-size:24px"> Example </primary></h3>
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<ul>
<li><strong>Ex:-</strong> $3 \times 3$</li>
<li>$
\begin{aligned}& 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) \therefore A^{\top}=\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) \\ & |A T|=a_{11}\left(a_{22} \cdot a_{33}-a_{32} \cdot a_{23}\right) \text { - } a_{21}\left(a_{12} \cdot a_{33}-a_{13} a_{32}\right) +a_{31}\left(a_{12} \cdot a_{23}-a_{22} a_{13}\right) \\ & =a_{11} a_{22} a_{33}-a_{11} a_{32} a_{23} -a_{21} a_{12} a_{33}+a_{21} a_{13} a_{32} +a_{31} a_{12} a_{23}-a_{31} a_{22} a_{13} \end{aligned}
$</li>
</ul>
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<footer style = "font-size: 18px">Determinants  along the $2^{nd}$ Column $\rightarrow$ Properties of Determinant-1 $\rightarrow$ <span style = "color:blue"> Example </span> $\rightarrow$ Property 2 $\rightarrow$ Property 3 </footer>

<h3 id="success-stylefont-size25px-determinants-l-6-success-13"><Success style="font-size:25px"> Determinants L-6 </Success></h3>
<h3 id="primary-stylefont-size24px-property-2-primary"><primary style="font-size:24px"> Property 2 </primary></h3>
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<ul>
<li>
<p>Determinant of a diagonal matrix is the product of its diagonal elements</p>
</li>
<li>
<p>$A=\left(\begin{array}{ccc}a_{11} & 0 & 0 \\ 0 & a_{22} & 0 \\ 0 & 0 & a_{33}\end{array}\right)$</p>
</li>
<li>
<p>$ |A| = a_{1 1} (a_{2 2} a_{3 3} - 0) - 0() + 0() $</p>
</li>
<li>
<p>$ = a_{1 1} a_{2 2} a{3 3} $</p>
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</li>
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<footer style = "font-size: 18px">Properties of Determinant-1 Column $\rightarrow$ Example $\rightarrow$ <span style = "color:blue"> Property 2 </span> $\rightarrow$ Property 3 $\rightarrow$ Property 4 </footer>

<h3 id="success-stylefont-size25px-determinants-l-6-success-14"><Success style="font-size:25px"> Determinants L-6 </Success></h3>
<h3 id="primary-stylefont-size24px-property-3-primary"><primary style="font-size:24px"> Property 3 </primary></h3>
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<div style='float: left; width:99%;font-size:30px; text-align:left;'>
<ul>
<li>
<p>The determinant of a triangular matrix in the product of its diagonal elements</p>
</li>
<li>
<p>$A=\left[\begin{array}{ccc}a_{11} & a_{12} & a_{13} \\ 0 & a_{22} & a_{23} \\ 0 & 0 & a_{33}\end{array}\right]$</p>
</li>
<li>
<p>$ |A| = a_{1 1} (a_{2 2} a_{3 3} - 0.a_{ 2 3}) - 0 () + 0 () $</p>
</li>
<li>
<p>$ = a_{1 1} a_{2 2} a_{3 3} $.</p>
</li>
</ul>
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<footer style = "font-size: 18px">Example $\rightarrow$ Property 2 $\rightarrow$ <span style = "color:blue"> Property 3 </span> $\rightarrow$ Property 4 $\rightarrow$ Property 4 </footer>

<h3 id="success-stylefont-size25px-determinants-l-6-success-15"><Success style="font-size:25px"> Determinants L-6 </Success></h3>
<h3 id="primary-stylefont-size24px-property-4-primary"><primary style="font-size:24px"> Property 4 </primary></h3>
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<div style='float: left; width:99%;font-size:30px; text-align:left;'>
<ul>
<li>
<p>If we multiply each element of a row (or column) by a constant k then the determinant will also be multiplied by k</p>
</li>
<li>
<p>i.e; if $A=\left[\begin{array}{ccc}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}\end{array}\right]$</p>
</li>
<li>
<p>$B=\left[\begin{array}{ccc}k a_{11} & k a_{12} & k a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}\end{array}\right]$</p>
</li>
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<footer style = "font-size: 18px">Property 2 $\rightarrow$ Property 3 $\rightarrow$ <span style = "color:blue"> Property 4 </span> $\rightarrow$ Property 4 $\rightarrow$ Thank you </footer>

<h3 id="success-stylefont-size25px-determinants-l-6-success-16"><Success style="font-size:25px"> Determinants L-6 </Success></h3>
<h3 id="primary-stylefont-size24px-property-4-primary-1"><primary style="font-size:24px"> Property 4 </primary></h3>
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<ul>
<li>
<p>Then |B| is</p>
</li>
<li>
<p>$ k a_{1 1} (a_{2 2} a_{2 3} - a_{2 3} a_{3 2}) $</p>
</li>
<li>
<p>$ - k a_{1 2} (a_{2 1} a_{3 3} - a_{3 1} a_{2 3}) $</p>
</li>
<li>
<p>$ + k a_{1 3} (a_{2 1} a_{3 2} - a_{2 2} a_{3 1}) $</p>
</li>
<li>
<p>$ = k (a_{1 1} (a_{2 2} a_{2 3} - a_{2 3} a_{3 2}) - a_{1 2} (a_{2 1} a_{3 3} - a_{3 1} a_{2 3}) + a_{1 3} (a_{2 1} a_{3 2} - a_{2 2} a_{3 1}))$</p>
</li>
<li>
<p>$ = k |A| $</p>
</li>
</ul>
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<footer style = "font-size: 18px">Property 3 $\rightarrow$ Property 4 $\rightarrow$ <span style = "color:blue"> Property 4 </span> $\rightarrow$ Thank you $\rightarrow$  </footer>