Question

What is the value of x if, \(\begin{vmatrix}x & 3 & 6 \\3 & 6 & x \\6 & x & 3 \end {vmatrix}\) = \(\begin{vmatrix}2 & x & 7 \\x & 7 & 2 \\7 & 2 & x \end {vmatrix}\) = \(\begin{vmatrix}4 & 5 & x \\5 & x & 4 \\x & 4 & 6 \end {vmatrix}\)?

(a) 9

(b) -9

(c) 0

(d) Can’t be predicted

The question was asked in an interview.

My doubt is from Application of Determinants topic in section Determinants of Mathematics – Class 12

Answer 1

Right answer is (b) -9

Easy explanation: Given that,

\(\begin{vmatrix}x & 3 & 6 \\3 & 6 & x \\6 & x & 3 \end {vmatrix}\) = \(\begin{vmatrix}2 & x & 7 \\x & 7 & 2\\7 & 2 & x \end {vmatrix}\) = \(\begin{vmatrix}4 & 5 & x \\ 5 & x & 4 \\x & 4 & 6 \end {vmatrix}\)

So, by circular determinant property,

Sum of the elements of a row = 0

So, x + 3 + 6 = 2 + x + 7 = 4 + 5 + x = 0

=> x = -9