Prove that \(\begin{vmatrix} 1 &x^2 &x^3 \\ 1 & y^2 &y^3 \\1 &z^2 &z^3 \end{vmatrix}\) = (x - y)(y - z)(z - x)(xy + yz + zx).

In a triangle ABC, if \(\begin{vmatrix} 1& 1 &1 \\1+sin A &1+sin B &1+sin C \\ sinA(1+sin A) &sin B(1+sin B) &sin C(1+sin C) \end{vmatrix}=0,\)
prove that \(\triangle\)ABC is an isosceles triangle.

Solve the following problems by using Factor Theorem :
Show that \(\begin{vmatrix}x & a & a \\ a & x & a \\ a &a & x \end{vmatrix}=(x-a)^2(x+2a)\)

Solve the following problems by using Factor Theorem :
Show that \(\begin{vmatrix} b+c & a-c & a-b \\ b-c & c+a & b-a \\ c-b & c-a & a+b \end{vmatrix}=8abc\)

Solve the following problems by using Factor Theorem :
Solve \(\begin{vmatrix} x+a &b &c \\ a & x+b & c \\ a & b &x+c \end{vmatrix}=0\)

Solve the following problems by using Factor Theorem :
Show that \(\begin{vmatrix} b+c & a &a^2 \\ c+a &b &b^2 \\ a+b & c & c^2 \end{vmatrix}\) = (a + b + c)(a - b)(b - c)(c - a).

Solve the following problems by using Factor Theorem :
Solve \(\begin{vmatrix} 4-x & 4+x & 4+x \\ 4+x & 4-x & 4+x \\ 4+x & 4+x & 4-x \end{vmatrix}=0\) .

Show that \(\begin{vmatrix} 1 &1 &1 \\ x & y & z \\ x^2 & y^2 & z^2 \end{vmatrix}\) = (x - y)( y - z)(z - x).

If A_i, B_i, C_i are the cofactors of a_i, b_i,c_i, respectively, i = 1 to 3 in
|A| = \(\begin{vmatrix} a_1 &b_1 &c_1 \\ a_2 & b_2 &c_2 \\ a_3 & b_3 & c_3 \end{vmatrix}\), show that \(\begin{vmatrix} A_1 &B_1 &C_1 \\ A_2 & B_2 &C_2 \\ A_3 & B_3 & C_3 \end{vmatrix}\) = |A|²