Using properties of determinants prove the following:
$∣ ∣ ∣ ∣ \begin{matrix} 1 & 1 & 1 a & b & c a^{3} & b^{3} & c^{3} \end{matrix} ∣ ∣ ∣ ∣ = (a - b) (b - c) (c - a) (a + b + c)$

Question

Using properties of determinants prove that -begin-vmatrix- -b-c- 2 - - a- 2 - -  -c-a- 2 - - b- 2 - - ca  -a-b- 2 - - c- 2 - - ab end-vmatrix-a- 2 -b- 2 -c- 2 -a-b-b-c-c-a-a-b-c-

Toppr Admin · Accepted Answer

Using properties of determinants prove that :$$\begin{vmatrix} (b+c)^{ 2 } & a^{ 2 } & bc \\ (c+a)^{ 2 } & b^{ 2 } & ca \\ (a+b)^{ 2 } & c^{ 2 } & ab \end{vmatrix}=(a^{ 2 }+b^{ 2 }+c^{ 2 })(a-b)(b-c)(c-a)(a+b+c)$$

Using properties of determinants prove that :
$$\begin{vmatrix} (b+c)^{ 2 } & a^{ 2 } & bc \\ (c+a)^{ 2 } & b^{ 2 } & ca \\ (a+b)^{ 2 } & c^{ 2 } & ab \end{vmatrix}=(a^{ 2 }+b^{ 2 }+c^{ 2 })(a-b)(b-c)(c-a)(a+b+c)$$