Consider a system of equations given below:
$ p_1 + p_2 + p_3 + p_4 + p_5 = 1 $
$ x_1*p_1 + x_2*p_2 + x_3*p_3 +x_4*p_4 =0$
$ x_1^2*p_1 + x_2^2*p_2 + x_3^2*p_3 +x_4^2*p_4 =1$
$ x_1^3*p_1 + x_2^3*p_2 + x_3^3*p_3 +x_4^3*p_4 =v_1$
$ x_1^4*p_1 + x_2^4*p_2 + x_3^4*p_3 +x_4^4*p_4 =v_2$
$ x_1^5*p_1 + x_2^5*p_2 + x_3^5*p_3 +x_4^5*p_4 =v_3$
$ x_1^6*p_1 + x_2^6*p_2 + x_3^6*p_3 +x_4^6*p_4 =v_4$
$ x_1^7*p_1 + x_2^7*p_2 + x_3^7*p_3 +x_4^7*p_4 =v_5$
$ x_1^8*p_1 + x_2^8*p_2 + x_3^8*p_3 +x_4^8*p_4 =v_6$
Is there a method to solve $x_i$ and $p_i$ in terms of $v_i$?
Thank You!
Hint:
A nxn nonhomogeneous system of linear equations has a unique non-trivial solution if and only if its determinant is non-zero. If this determinant is zero, then the system has either no nontrivial solutions or an infinite number of solutions.
Use matrices to represent this system of equations. Get the determinant of the coefficient matrix.