Question

What will be the value of \(\begin{vmatrix}a & b & c \\b & c & a \\c & a & b \end {vmatrix}\)?

(a) (a^3 + b^3 + c^3 + 3abc)

(b) –(a^3 + b^3 + c^3 + 3abc)

(c) (a^3 + b^3 + c^3 – 3abc)

(d) –(a^3 + b^3 + c^3 – 3abc)

I got this question during an interview.

This question is from Determinant topic in chapter Determinants of Mathematics – Class 12

Answer 1

Right option is (d) –(a^3 + b^3 + c^3 – 3abc)

The explanation: Given, \(\begin{vmatrix}a & b & c \\b & c & a \\c & a & b \end {vmatrix}\)

Replacing R1 = R1 + R2 + R3

\(\begin{vmatrix}a + b + c & a + b + c & a + b + c \\b & c & a \\c & a & b \end {vmatrix}\)

= (a + b + c)\(\begin{vmatrix}1 & 1 & 1 \\b & c & a \\c & a & b \end {vmatrix}\)

Replacing 2^nd column by C2 – C1 and 3^rd column by C3 – C1

= (a + b + c)\(\begin{vmatrix}1 & 0 & 0 \\b & c-b & a-b \\c & a-c & b-c \end {vmatrix}\)

= (a + b + c)[(c – b)(b – c) – (a – b)(a – c)]

= (a + b + c)(bc – b^2 – c^2 + bc + a^2 + ac + ab – bc)

= -(a + b + c)(a^2 + b^2 + c^2 – ab – bc – ac)

= -(a^3 + b^3 + c^3 – 3abc)