First slide

Introduction to Determinants

Question

${If a}_{1}, a_{2,} a_{3}, 5, 4, a_{6,} a_{7,} a_{8,} a_{9} a r e i n H . P . a n d t h e v a l u e o f t h e d e t e r m i n a n t$
$_{|\begin{matrix} a_{1} & a_{2} & a_{3} \\ 5 & 4 & a_{6} \\ a_{7} & a_{8} & a_{9} \end{matrix}| i s D, t h e n t h e v a l u e o f 21 D i s}$

Moderate

Solution

${We have a}_{1}, a_{2,} a_{3}, 5, 4, a_{6,} a_{7,} a_{8,} a_{9} a r e i n H . P . w e g e t a_{n} = \frac{20}{n} d = \frac{1}{20} a n d |\begin{matrix} a_{1} & a_{2} & a_{3} \\ 5 & 4 & a_{6} \\ a_{7} & a_{8} & a_{9} \end{matrix}| = D$

$D = |\begin{matrix} 20 & \frac{20}{2} & \frac{20}{3} \\ \frac{20}{4} & \frac{20}{5} & \frac{20}{6} \\ \frac{20}{7} & \frac{20}{8} & \frac{20_{}}{9} \end{matrix}|$
$(\sin c e \frac{1}{a_{1}}, \frac{1}{a_{2},} \frac{1}{a_{3_{}}}, \frac{1}{5}, \frac{1}{4}, . . . ., \frac{1}{a_{9}} a r e i n A P \Rightarrow c o m m o n d i f f e r e n c e = \frac{1}{4} - \frac{1}{5} = \frac{1}{20} a n d 4 t h t e r m = \frac{1}{5} = \frac{1}{a_{1}} + \frac{(3)}{20} \Rightarrow a_{1} = 20$
$= \frac{{(20)}^{3}}{4 x 7} |\begin{matrix} 1 & \frac{1}{2} & \frac{1}{3} \\ 1 & \frac{4}{5} & \frac{2}{3} \\ 1 & \frac{7}{8} & \frac{7}{9} \end{matrix}| A p p l y i n g R_{1} \to R_{1} - R_{2,} R_{2} \to R_{2} - R_{3,}$

$= \frac{{(20)}^{3}}{4 x 7} |\begin{matrix} 0 & \frac{- 3}{10} & \frac{- 1}{3} \\ 0 & \frac{- 3}{40} & \frac{- 1}{9} \\ 1 & \frac{7}{8} & \frac{7}{9} \end{matrix}| = \frac{50}{21} \Rightarrow 21 D = 50$

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