Using properties of determinants-prove that - beginvmatrixx-y - x - x 5 x-4 y - 4 x - 2 x10 x-8 y - 8 x - 3 x

X+y amp;x nbsp; amp;x nbsp; nbsp;5x+4y amp; nbsp;4x amp; 2x nbsp; nbsp;10x+8y amp; 8x amp;3x nbsp; , R_3→ R_3 2R_2 nbsp; ...

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Byju's Answer

Standard XII

Mathematics

Special Determinants

Using propert...

Question

Using properties of determinants,prove that : $∣ ∣ ∣ ∣ \begin{matrix} x + y & x & x 5 x + 4 y & 4 x & 2 x 10 x + 8 y & 8 x & 3 x \end{matrix} ∣ ∣ ∣ ∣ = x^{3}$

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Solution

$∣ ∣ ∣ ∣ \begin{matrix} x + y & x & x 5 x + 4 y & 4 x & 2 x 10 x + 8 y & 8 x & 3 x \end{matrix} ∣ ∣ ∣ ∣, \begin{matrix} R_{3} \to R_{3} - 2 R_{2} R_{2} \to R_{2} - 4 R_{1} \end{matrix}$

$= ∣ ∣ ∣ ∣ \begin{matrix} x + y & x & x x & 0 & - 2 x 0 & 0 & - x \end{matrix} ∣ ∣ ∣ ∣, expand along R_{3}$

$= - x ∣ ∣ ∣ \begin{matrix} x + y & x x & 0 \end{matrix} ∣ ∣ ∣ = - x (0 - x^{2}) = x^{3}$

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Using properties of determinants,prove that : ∣∣ ∣∣x+yxx5x+4y4x2x10x+8y8x3x∣∣ ∣∣=x3

∣∣ ∣∣x+yxx5x+4y4x2x10x+8y8x3x∣∣ ∣∣,R3→R3−2R2R2→R2−4R1 =∣∣ ∣∣x+yxxx0−2x00−x∣∣ ∣∣,expand along R3 =−x∣∣∣x+yxx0∣∣∣=−x(0−x2)=x3

Using properties of determinants,prove that : $∣ ∣ ∣ ∣ \begin{matrix} x + y & x & x 5 x + 4 y & 4 x & 2 x 10 x + 8 y & 8 x & 3 x \end{matrix} ∣ ∣ ∣ ∣ = x^{3}$