If p - q - r - a - b - c - 0- then the determinant pa qb rc qc ra pb rb pc qa equals-

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Definition of a Determinant

If p + q + ...

Question

If $p + q + r = a + b + c = 0$ , then the determinant $∣ ∣ ∣ ∣ \begin{matrix} p a & q b & r c q c & r a & p b r b & p c & q a \end{matrix} ∣ ∣ ∣ ∣$ equals.

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p a + q b + r c

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p a + q b + r c + a + b + c

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Solution

The correct option is D $0$
determinant of matrix $= p a (q r a^{2} - p^{2} b c) - q b (q^{2} c a - p r b^{2}) + r c (p q c^{2} - r^{2} a b)$

$= p q r (a^{3} + b^{3} + c^{3}) - a b c (p^{3} + q^{3} + r^{3})$

$= p q r (3 a b c) - a b c (3 p q r)$ ......(as when $a + b + c = 0, a^{3} + b^{3} + c^{3} = 3 a b c$ )

$= 0$

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Similar questions

If p + q + r = a + b + c = 0, then the determinant
$Δ = ∣ ∣ ∣ ∣ \begin{matrix} p a & q b & r c q c & r a & p b r b & p c & q a \end{matrix} ∣ ∣ ∣ ∣$ equals

Q. If

p + q + r = a + b + c = 0

then the determinant

Δ = ∣ ∣ ∣ ∣ \begin{matrix} p a & q b & r c q c & r a & p b r b & p c & q a \end{matrix} ∣ ∣ ∣ ∣

equals

Q. If

p + q + r = 0 = a + b + c,

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Q. If

p + q + r = 0 = a + b + c,

then the value of the determinant

∣ ∣ ∣ ∣ \begin{matrix} p a & q b & r c q c & r a & p b r b & p c & q a \end{matrix} ∣ ∣ ∣ ∣

Q. If

p + q + r = 0 = a + b + c

, then the value of the determinant

∣ ∣ ∣ ∣ \begin{matrix} p a & q b & r c q c & r a & p b r b & p c & q a \end{matrix} ∣ ∣ ∣ ∣

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If p+q+r=a+b+c=0, then the determinant ∣∣ ∣∣paqbrcqcrapbrbpcqa∣∣ ∣∣ equals.

The correct option is D 0determinant of matrix =pa(qra2−p2bc)−qb(q2ca−prb2)+rc(pqc2−r2ab)=pqr(a3+b3+c3)−abc(p3+q3+r3)=pqr(3abc)−abc(3pqr) ......(as when a+b+c=0,a3+b3+c3=3abc)=0

If $p + q + r = a + b + c = 0$ , then the determinant $∣ ∣ ∣ ∣ \begin{matrix} p a & q b & r c q c & r a & p b r b & p c & q a \end{matrix} ∣ ∣ ∣ ∣$ equals.