Question

What is the relation between the two determinants f(x) = \(\begin{vmatrix}–a^2 & ab & ac \\ab & -b^2 & bc \\ac & bc & -c^2 \end {vmatrix}\) and g(x) = \(\begin{vmatrix}0 & c & b \\c & 0 & a \\b & a & 0 \end {vmatrix}\)?

(a) f(x) = g(x)

(b) f(x) = (g(x))^2

(c) g(x) = 2f(x)

(d) g(x) = (f(x))^2

The question was asked in final exam.

Question is from Application of Determinants in portion Determinants of Mathematics – Class 12

Answer 1

Correct answer is (b) f(x) = (g(x))^2

Easiest explanation: Let, D = \(\begin{vmatrix}0 & c & b \\c & 0 & a \\b & a & 0 \end {vmatrix}\)

Expanding D by the 1^st row we get,

D = – c\(\begin{vmatrix}c & a \\b & 0 \end {vmatrix}\) + b\(\begin{vmatrix}c & 0 \\b & a \end {vmatrix}\)

= – c(0 – ab) + b(ac – 0)

= 2abc

Now, we have adjoint of D = D’

= \(\begin{vmatrix}

\begin{vmatrix}0 & a\\ a & 0\\ \end{vmatrix} & – \begin{vmatrix}c & a\\ b & 0\\ \end{vmatrix} & \begin{vmatrix}c & 0\\ b & a\\ \end{vmatrix}\\

– \begin{vmatrix}c & b\\ a & 0\\ \end{vmatrix} & \begin{vmatrix}0 & b\\ b & 0\\ \end{vmatrix} & – \begin{vmatrix}0 & c\\ b & 0\\ \end{vmatrix} \\

\begin{vmatrix}c & b\\ 0 & a\\ \end{vmatrix} & – \begin{vmatrix}0 & b\\ c & a\\ \end{vmatrix} & \begin{vmatrix}0 & c\\ c & 0\\ \end{vmatrix} \\

\end{vmatrix}\)   

Or, D’ = \(\begin{vmatrix}–a^2 & ab & ac \\ab & -b^2 & bc \\ac & bc & -c^2 \end {vmatrix}\)

Or, D’ = D^2

Or, D’ = D^2 = (2abc)^2