Question

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Answer 1

Question :- Find the value of determinant

$\begin{gathered}\sf \left | \begin{array}{ccc}4&a& {a}^{2} \\4&b& {b}^{2} \\4&c& {c}^{2} \end{array}\right | \end{gathered}$

$\green{\large\underline{\sf{Solution-}}}$

Given determinant is

$\begin{gathered}\sf \left | \begin{array}{ccc}4&a& {a}^{2} \\4&b& {b}^{2} \\4&c& {c}^{2} \end{array}\right | \end{gathered}$

Take out 4 common from first column, we get

$\rm \: = \: 4\begin{gathered}\sf \left | \begin{array}{ccc}1&a& {a}^{2} \\1&b& {b}^{2} \\1&c& {c}^{2} \end{array}\right | \end{gathered}$

$\: \: \: \: \: \: \: \: \: \: \: \red{ \boxed{ \sf{ \:R_2 \: \to \: R_2 - R_1 \: }}}$

$\rm \: = \: 4\begin{gathered}\sf \left | \begin{array}{ccc}1&a& {a}^{2} \\0&b - a& {b}^{2} - {a}^{2} \\1&c& {c}^{2} \end{array}\right | \end{gathered}$

$\: \: \: \: \: \: \: \: \: \: \: \red{ \boxed{ \sf{ \:R_3 \: \to \: R_3 - R_1 \: }}}$

$\rm \: = \: 4\begin{gathered}\sf \left | \begin{array}{ccc}1&a& {a}^{2} \\0&b - a& {b}^{2} - {a}^{2} \\0&c - a& {c}^{2} - {a}^{2} \end{array}\right | \end{gathered}$

$\rm \: = \: 4\begin{gathered}\sf \left | \begin{array}{ccc}1&a& {a}^{2} \\0&b - a& (b - a)(b + a) \\0&c - a& (c - a)(c + a)\end{array}\right | \end{gathered}$

Take out b - a and c - a common from Row 2 and Row 3, we get

$\rm \: = \: 4(b - a)(c - a)\begin{gathered}\sf \left | \begin{array}{ccc}1&a& {a}^{2} \\0&1& b + a \\0&1& c + a\end{array}\right | \end{gathered}$

$\: \: \: \: \: \: \: \: \: \: \: \red{ \boxed{ \sf{ \:R_3 \: \to \: R_3 - R_2 \: }}}$

$\rm \: = \: 4(b - a)(c - a)\begin{gathered}\sf \left | \begin{array}{ccc}1&a& {a}^{2} \\0&1& b + a \\0&0& c - b\end{array}\right | \end{gathered}$

On expanding along column 1, we get

$\rm \: = \: 4(b - a)(c - a)(c - b)$

can be rewritten as

$\rm \: = \: 4(a - b)(c - a)(b - c)$

Hence,

$\rm\implies \:\boxed{ \rm{ \:\begin{gathered}\sf \left | \begin{array}{ccc}4&a& {a}^{2} \\4&b& {b}^{2} \\4&c& {c}^{2} \end{array}\right | \end{gathered} = 4(a - b)(b - c)(c - a) \: }}$

$\rule{190pt}{2pt}$

Additional Information :-

1. The determinant value remains unaltered if rows and columns are interchanged.

2. The determinant value is 0, if two rows or columns are identical.

3. The determinant value is multiplied by - 1, if successive rows or columns are interchanged.

4. The determinant value remains unaltered if rows or columns are added or subtracted.