Without expanding at any stage- find the value of a b c a-2x b-2y c-2z x y z-

Apply R_2→ R_2 2R_3 D= a amp; b amp; c a +2x 2x amp; b +2x 2x amp; c+2x 2x x amp; y amp; z D= a amp; b amp; c ...

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Byju's Answer

Standard XII

Mathematics

Properties of Determinants

Without expan...

Question

Without expanding at any stage, find the value of $∣ ∣ ∣ ∣ \begin{matrix} a & b & c a + 2 x & b + 2 y & c + 2 z x & y & z \end{matrix} ∣ ∣ ∣ ∣$ .

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Solution

Apply $R_{2} \to R_{2} - 2 R_{3}$
$D = ∣ ∣ ∣ ∣ \begin{matrix} a & b & c a + 2 x - 2 x & b + 2 x - 2 x & c + 2 x - 2 x x & y & z \end{matrix} ∣ ∣ ∣ ∣$
$D = ∣ ∣ ∣ ∣ \begin{matrix} a & b & c a & b & c x & y & z \end{matrix} ∣ ∣ ∣ ∣$
Now as $R_{1}$ and $R_{2}$ are same $∴ D = 0$ .

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Without expanding at any stage, find the value of ∣∣ ∣∣abca+2xb+2yc+2zxyz∣∣ ∣∣.

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Without expanding at any stage, find the value of $∣ ∣ ∣ ∣ \begin{matrix} a & b & c a + 2 x & b + 2 y & c + 2 z x & y & z \end{matrix} ∣ ∣ ∣ ∣$ .