By using properties of determinants- show that- a2-1 a b a c- a b b2-1 b c- c a c b c2-1 -1-a2-b2-c2

Taking out common factors a, b, and c from R1, R2, and R3 respectively, we have: Applying R2 → R2 − R1 and R3 → R3 − R1, we have: Applying C1 → aC1, C2 → bC2 ...

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Standard XII

Mathematics

Invertible Element Binary Operation

By using prop...

Question

By using properties of determinants, show that:

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Solution

Taking out common factors a, b, and c from R₁, R₂, and R₃respectively, we have:

Applying R₂ → R₂ − R₁and R₃ → R₃ − R₁, we have:

Applying C₁ → aC₁, C₂→ bC_2,and C₃ → cC₃, we have:

Expanding along R₃, we have:

Hence, the given result is proved.

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By using properties of determinants, show that:

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∣ ∣ ∣ ∣ ∣ \begin{matrix} a^{2} + 1 & a b & a c a b & b^{2} + 1 & b c c a & c b & c^{2} + 1 \end{matrix} ∣ ∣ ∣ ∣ ∣ = 1 + a^{2} + b^{2} + c^{2}

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By using properties of determinants, show that:

Taking out common factors a, b, and c from R1, R2, and R3 respectively, we have: Applying R2 → R2 − R1 and R3 → R3 − R1, we have: Applying C1 → aC1, C2 → bC2, and C3 → cC3, we have: Expanding along R3, we have: Hence, the given result is proved.

Taking out common factors a, b, and c from R₁, R₂, and R₃respectively, we have:

Applying R₂ → R₂ − R₁and R₃ → R₃ − R₁, we have:

Applying C₁ → aC₁, C₂→ bC_2,and C₃ → cC₃, we have:

Expanding along R₃, we have:

Hence, the given result is proved.