SATHEE: Matrices and Determinants 2 Question 29

Matrices and Determinants 2 Question 29

32. The determinants $| \begin{matrix} 1 & a & b c \\ 1 & b & c a \\ 1 & c & a b \end{matrix} |$ and $| \begin{matrix} 1 & a & a^{2} \\ 1 & b & b^{2} \\ 1 & c & c^{2} \end{matrix} |$ are not identically equal.

(1983, 1M)

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Answer:

Correct Answer: 32. False

Solution:

Let $Δ = | \begin{matrix} 1 & a & b c \\ 1 & b & c a \\ 1 & c & a b \end{matrix} | = \frac{1}{a b c} | \begin{matrix} a & a^{2} & a b c \\ b & b^{2} & a b c \\ c & c^{2} & a b c \end{matrix} |$

Applying $R_{1} \to a R_{1}, R_{2} \to b R_{2}, R_{3} \to c R_{3}$

$= \frac{1}{a b c} \cdot a b c$ $| \begin{matrix} a & a^{2} & 1 \\ b & b^{2} & 1 \\ c & c^{2} & 1 \end{matrix} |$ $= | \begin{matrix} 1 & a & a^{2} \\ 1 & b & b^{2} \\ 1 & c & c^{2} \end{matrix} |$

$∴ | \begin{matrix} 1 & a & b c \\ 1 & b & c a \\ 1 & c & a b \end{matrix} |$ $= | \begin{matrix} 1 & a & a^{2} \\ 1 & b & b^{2} \\ 1 & c & c^{2} \end{matrix} |$

Hence, statement is false.

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Matrices and Determinants 2 Question 29

32. The determinants |1abc1bca1cab| and |1aa21bb21cc2| are not identically equal.

Answer:

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32. The determinants $| \begin{matrix} 1 & a & b c \\ 1 & b & c a \\ 1 & c & a b \end{matrix} |$ and $| \begin{matrix} 1 & a & a^{2} \\ 1 & b & b^{2} \\ 1 & c & c^{2} \end{matrix} |$ are not identically equal.