Question

If, Si = a^i + b^i + c^i then what is the value of \(\begin{vmatrix}S0 & S1 & S2 \\S1 & S2 & S3 \\S2 & S3 & S4 \end {vmatrix}\)?

(a) (a + b)^2(b – c)^2(c – a)^2

(b) (a – b)^2(b – c)^2(c + a)^2

(c) (a – b)^2(b – c)^2(c – a)^2

(d) (a – b)^2(b + c)^2(c – a)^2

The question was posed to me in an interview for job.

Asked question is from Application of Determinants topic in portion Determinants of Mathematics – Class 12

Answer 1

Right choice is (c) (a – b)^2(b – c)^2(c – a)^2

The best explanation: We have, \(\begin{vmatrix}1 & 1 & 1 \\a & b & c \\a^2 & b^2 & c^2 \end {vmatrix}\)

So, the value of the \(\begin{vmatrix}1 & 1 & 1 \\a & b & c \\a^2 & b^2 & c^2 \end {vmatrix}\) = (a – b)(b – c)(c – a)

Now, by circulant determinant,

\(\begin{vmatrix}1 & 1 & 1 \\a & b & c \\a^2 & b^2 & c^2 \end {vmatrix}\) X \(\begin{vmatrix}1 & 1 & 1 \\a & b & c \\a^2 & b^2 & c^2 \end {vmatrix}\) = \(\begin{vmatrix}S0 & S1 & S2 \\S1 & S2 & S3 \\S2 & S3 & S4 \end {vmatrix}\)

Multiplying the determinant in row by row,

We get, (a – b)^2(b – c)^2(c – a)^2