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CBSE Class 12 · 2013 · DeterminantsLong Answer

Using properties of determinants, prove the following:

1 x^{2} x x 1 x^{2} x^{2} x 1 = (1 - x^{3})^{2} .

Answer: Proved

Solution

Apply

C_{1} \to C_{1} + C_{2} + C_{3}

; each entry of column

1

becomes

1 + x + x^{2}

. Take it common:

= (1 + x + x^{2}) 111 x 1 x^{2} x^{2} x 1 .

Apply

R_{1} \to R_{1} - R_{2}

,

R_{2} \to R_{2} - R_{3}

, take

(x - 1)

factors and expand:

= (1 + x + x^{2}) (x - 1)^{2} (1 + x + x^{2}) = (x^{3} - 1)^{2} = (1 - x^{3})^{2} .

Proved.

CBSE Class 12 Board · 2013 · Subjective

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