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:

  1. 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.



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