Prove the determinant of this matrix linear algebra Use prop

Prove the determinant of this matrix.

linear algebra

Use properties of determinants to deduce that det [1 1 1 1 a b c d a^2 b^2 c^2 d^2 a^3 b^3 c^3 d^3] = (b - a)(c - a)(d - a)(c - b)(d - b)(d - c)

Solution

In the given determinant, if we replace b by a, the the 1st and the 2nd columns are identical so that the determinant\'s value becomes 0. Hence (b-a) must be a factor of the expression for the value of the determinant. Similarly, (c-a), (d-a), ( c-b), (d-b),and (d-c) are factors of the expression for the value of the determinant. Further, the combined degree of any term in the expression for the value of the determinant is 6. Also, the combined degree of any term in the expansion of (b-a)(c-a)(d-a)(c-b)(d-b)(d-c) is 6. Therefore, the given determinant = ± (b-a)(c-a)(d-a)(c-b)(d-b)(d-c), since the coefficient of anty term in the expansion of (b-a)(c-a)(d-a)(c-b)(d-b)(d-c) is ±1 . On comparison of any term in the expression for the value of the determinant and the expansion of (b-a)(c-a)(d-a)(c-b)(d-b)(d-c),(say of bc²d³, we find that the value of the given determinant=the expansion of (b-a)(c-a)(d-a)(c-b)(d-b)(d-c)