By using properties of determinants- show that-i - 1 a a2- 1 b b2- 1 c c2 -a-bb-cc-aii - 1 1 1- a b c- a3 b3 c3 -a-bb-cc-aa-b-c

(i) Applying R1 → R1 − R3 and R2 → R2 − R3, we have: Applying R1 → R1 + R2, we have: Expanding along C1, we have: Hence, the given result is proved ...

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Standard XII

Mathematics

Differentiation under Integral Sign

By using prop...

Question

By using properties of determinants, show that:

(i)

(ii)

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Solution

(i)

Applying R₁ → R₁ − R₃and R₂ → R₂ − R₃, we have:

Applying R₁ → R₁ + R₂, we have:

Expanding along C₁, we have:

Hence, the given result is proved.

(ii) Let.

Applying C₁ → C₁ − C₃and C₂ → C₂ − C₃, we have:

Applying C₁ → C₁ + C₂, we have:

Expanding along C₁, we have:

Hence, the given result is proved.

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Q. Using the properties of determinants, show that:
(i)

∣ ∣ ∣ ∣ ∣ \begin{matrix} 1 & a & a^{2} 1 & b & b^{2} 1 & c & c^{2} \end{matrix} ∣ ∣ ∣ ∣ ∣ = (a - b) (b - c) (c - a)

(ii)

∣ ∣ ∣ ∣ ∣ \begin{matrix} 1 & 1 & 1 a & b & c a^{3} & b^{3} & c^{3} \end{matrix} ∣ ∣ ∣ ∣ ∣ = (a - b) (b - c) (c - a) (a + b + c)

Q. Using properties of determinants, prove the following:

∣ ∣ ∣ ∣ \begin{matrix} a & b & c a - b & b - c & c - a b + c & c + a & a + b \end{matrix} ∣ ∣ ∣ ∣ = a^{3} + b^{3} + c^{3} - 3 a b c

Q. Using properties of determinants, prove the following:

∣ ∣ ∣ ∣ \begin{matrix} 1 & 1 & 1 a & b & c a^{3} & b^{3} & c^{3} \end{matrix} ∣ ∣ ∣ ∣ = (a - b) (b - c) (c - a) (a + b + c)

Q. Using properties of determinants prove the following:

∣ ∣ ∣ ∣ \begin{matrix} 1 & 1 & 1 a & b & c a^{3} & b^{3} & c^{3} \end{matrix} ∣ ∣ ∣ ∣ = (a - b) (b - c) (c - a) (a + b + c)

Using the property of determinants and without expanding, prove that:

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By using properties of determinants, show that: (i) (ii)

By using properties of determinants, show that:

(i)

(ii)