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Properties of Determinants

Using propert...

Question

Using properties of determinants, prove that
$⎛ ⎜ ⎝ \begin{matrix} b + c & a - b & a c + a & b - c & b a + b & c - a & c \end{matrix} ⎞ ⎟ ⎠$ = $3 a b c - a^{3} - b^{3} - c^{3}$

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Solution

$∣ ∣ ∣ ∣ \begin{matrix} b + c & a - b & a c + a & b - c & b a + b & c - a & c \end{matrix} ∣ ∣ ∣ ∣$
Put $a = 1; b = 1; c = 1$
$∣ ∣ ∣ ∣ \begin{matrix} 2 & 0 & 1 2 & 0 & 1 2 & 0 & 1 \end{matrix} ∣ ∣ ∣ ∣ = 0 = L . H . S R . H . S \Rightarrow 3 a b c - a^{3} - b^{3} - c^{3} \Rightarrow 3 - 3 = 0 ∴ R . H . S = L . H . S$
$∴$ Hence Proved.

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

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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. If a + b + c = 5 and ab + bc + ca = 10, prove that a³ + b³ + c³ − 3abc = −25.

Q. Question 8
If a + b + c = 5 and ab + bc + ca = 10, then prove that

a^{3} + b^{3} + c^{3} - 3 a b c = - 25

.

\frac{a^{3} + b^{3} + c^{3} - 3 a b c}{a^{2} + b^{2} + c^{2} - a b - b c - c a}

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