Question

If, x^3 = 1, then, what will be the value of\(\begin{vmatrix}a & b & c \\b & c & a \\c & a & b \end {vmatrix}\)?

(a) -(a + bx + cx^2)\(\begin{vmatrix}1 & b & c \\x^2 & c & a \\x & a & b \end {vmatrix}\)

(b) (a + bx + cx^2)\(\begin{vmatrix}1 & b & c \\x^2 & c & a \\x & a & b \end {vmatrix}\)

(c) (a – bx – cx^2)\(\begin{vmatrix}1 & b & c \\x^2 & c & a \\x & a & b \end {vmatrix}\)

(d) (a + bx – cx^2)\(\begin{vmatrix}1 & b & c \\x^2 & c & a \\x & a & b \end {vmatrix}\)

This question was posed to me in exam.

This interesting question is from Determinant in division Determinants of Mathematics – Class 12

Answer 1

The correct answer is (b) (a + bx + cx^2)\(\begin{vmatrix}1 & b & c \\x^2 & c & a \\x & a & b \end {vmatrix}\)

Easy explanation: We have, \(\begin{vmatrix}a & b & c \\b & c & a \\c & a & b \end {vmatrix}\)

As, x^3 = 1,

= \(\begin{vmatrix}a & bx & cx^2 \\b & cx & ax^2 \\c & ax & bx^2  \end {vmatrix}\)

Replacing the 1^st column by C1 + C2 + C3 we get,

= \(\begin{vmatrix}a + bx + cx^2 & bx & cx^2 \\ b + cx + ax^2 & cx & ax^2 \\c + ax + bx^2 & ax & bx^2  \end {vmatrix}\)

As, x^3 = 1 so, x^4 = x^3 * x = x

= \(\begin{vmatrix}a + bx + cx^2 & b & c \\x^2 (a + bx + cx^2) & c & a \\x(a + bx + cx^2) & a & b \end {vmatrix}\)

= (a + bx + cx^2)\(\begin{vmatrix}1 & b & c \\x^2 & c & a \\x & a & b \end {vmatrix}\)