Matrices and Determinants 2 Question 29
32. The determinants $\begin{vmatrix}1 & a & b c \\ 1 & b & c a \\ 1 & c & a b\end{vmatrix}$ and $\begin{vmatrix}1 & a & a^{2} \\ 1 & b & b^{2} \\ 1 & c & c^{2}\end{vmatrix}$ are not identically equal.
(1983, 1M)
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Answer:
Correct Answer: 32. False
Solution:
- Let $\Delta=\begin{vmatrix}1 & a & b c \\ 1 & b & c a \\ 1 & c & a b\end{vmatrix}=\frac{1}{a b c}\begin{vmatrix}a & a^{2} & a b c \\ b & b^{2} & a b c \\ c & c^{2} & a b c\end{vmatrix}$
Applying $R_{1} \rightarrow a R_{1}, R_{2} \rightarrow b R_{2}, R_{3} \rightarrow c R_{3}$
$ =\frac{1}{a b c} \cdot a b c$ $\begin{vmatrix} a & a^{2} & 1 \\ b & b^{2} & 1 \\ c & c^{2} & 1 \end{vmatrix} $ $=\begin{vmatrix} 1 & a & a^{2} \\ 1 & b & b^{2} \\ 1 & c & c^{2} \end{vmatrix} $
$\therefore \begin{vmatrix} 1 & a & b c \\ 1 & b & c a \\ 1 & c & a b \end{vmatrix} $ $=\begin{vmatrix} 1 & a & a^{2} \\ 1 & b & b^{2} \\ 1 & c & c^{2} \end{vmatrix}$
Hence, statement is false.
