Question

if

Δ_{1} = | \begin{array}{ccc} a_{1}^{2} + b_{1} + c_{1} & a_{1} a_{2} + b_{2} + c_{2} & a_{1} a_{3} + b_{3} + c_{3} \\ b_{1} b_{2} + c_{1} & b_{2}^{2} + c_{2} & b_{2} b_{3} + c_{3} \\ c_{3} c_{1} & c_{3} c_{2} & c_{3}^{2} \end{array} |

and

Δ_{2} = | \begin{array}{ccc} a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \end{array} |

then find the value of

\frac{Δ_{1}}{Δ_{2}}

.

Accepted Answer

$Δ_{1} = | \begin{array}{ccc} a_{1}^{2} + b_{1} + c_{1} & a_{1} a_{2} + b_{2} + c_{2} & a_{1} a_{3} + b_{3} + c_{3} \\ b_{1} b_{2} + c_{1} & b_{2}^{2} + c_{2} & b_{2} b_{3} + c_{3} \\ c_{3} c_{1} & c_{3} c_{2} & c_{3}^{2} \end{array} |$

$Δ_{1} = c_{3} | \begin{array}{ccc} a_{1}^{2} + b_{1} + c_{1} & a_{1} a_{2} + b_{2} + c_{2} & a_{1} a_{3} + b_{3} + c_{3} \\ b_{1} b_{2} + c_{1} & b_{2}^{2} + c_{2} & b_{2} b_{3} + c_{3} \\ c_{1} & c_{2} & c_{3} \end{array} |$

$= c_{3} | \begin{array}{ccc} a_{1}^{2} + b_{1} & a_{1} a_{2} + b_{2} & a_{1} a_{3} + b_{3} \\ b_{1} b_{2} + c_{1} & b_{2}^{2} + c_{2} & b_{2} b_{3} + c_{3} \\ c_{1} & c_{2} & c_{3} \end{array} | + | \begin{array}{ccc} c_{1} & c_{2} & c_{3} \\ b_{1} b_{2} + c_{1} & b_{2}^{2} + c_{2} & b_{2} b_{3} + c_{3} \\ c_{1} & c_{2} & c_{3} \end{array} |$

$= c_{3} | \begin{array}{ccc} a_{1}^{2} + b_{1} & a_{1} a_{2} + b_{2} & a_{1} a_{3} + b_{3} \\ b_{1} b_{2} + c_{1} & b_{2}^{2} + c_{2} & b_{2} b_{3} + c_{3} \\ c_{1} & c_{2} & c_{3} \end{array} | + 0$

$= c_{3} | \begin{array}{ccc} a_{1}^{2} + b_{1} & a_{1} a_{2} + b_{2} & a_{1} a_{3} + b_{3} \\ b_{1} b_{2} & b_{2}^{2} & b_{2} b_{3} \\ c_{1} & c_{2} & c_{3} \end{array} | + c_{3} | \begin{array}{ccc} a_{1}^{2} + b_{1} & a_{1} a_{2} + b_{2} & a_{1} a_{3} + b_{3} \\ c_{1} & c_{2} & c_{3} \\ c_{1} & c_{2} & c_{3} \end{array} |$

$= c_{3} b_{2} | \begin{array}{ccc} a_{1}^{2} + b_{1} & a_{1} a_{2} + b_{2} & a_{1} a_{3} + b_{3} \\ b_{1} & b_{2} & b_{3} \\ c_{1} & c_{2} & c_{3} \end{array} | + 0$

$= c_{3} b_{2} | \begin{array}{ccc} a_{1}^{2} & a_{1} a_{2} & a_{1} a_{3} \\ b_{1} & b_{2} & b_{3} \\ c_{1} & c_{2} & c_{3} \end{array} | + c_{3} b_{2} | \begin{array}{ccc} b_{1} & b_{2} & b_{3} \\ b_{1} & b_{2} & b_{3} \\ c_{1} & c_{2} & c_{3} \end{array} |$

$= a_{1} b_{2} c_{3} | \begin{array}{ccc} a_{1} & a_{2} & a_{3} \\ b_{1} & b_{2} & b_{3} \\ c_{1} & c_{2} & c_{3} \end{array} | + 0$

$= a_{1} b_{2} c_{3} ∆_{2}$

$\Rightarrow ∆_{1} = ∆_{2} a_{1} b_{2} c_{3}$

$\Rightarrow \frac{∆_{1}}{∆_{2}} = a_{1} b_{2} c_{3}$

Andhra Pradesh Board Solutions for Exercise 4: Exercise 3 (d)

Andhra Pradesh Board Mathematics Solutions for Exercise - Andhra Pradesh Board Solutions for Exercise 4: Exercise 3 (d)

Questions from Andhra Pradesh Board Solutions for Exercise 4: Exercise 3 (d) with Hints & Solutions

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