Consider the following statements : 1. The determinants $\left| \begin{matrix} 1 & a & bc \ 1 & b & ca \ 1 & c & ab \end{matrix} ight|$ and $\left| \begin{matrix} 1 & a & a^{2} \ 1 & b & b^{2} \ 1 & c & c^{2} \end{matrix} ight|$ are not identically equal. 2. For $a 0,\ b 0,\ c 0$ the value of the determinant $\left| \begin{matrix} a & b & c \ b & c & a \ c & a & b \end{matrix} ight|$ is always positive. 3.If$\ \left| \begin{matrix} x_{1} & y_{1} & 1 \ x_{2} & y_{2} & 1 \ x_{3} & y_{3} & 1 \end{matrix} ight| = \left| \begin{matrix} a_{1} & b_{1} & 1 \ a_{2} & b_{2} & 1 \ a_{3} & b_{3} & 1 \end{matrix} ight|$, then the two triangles with vertices $\left( x_{1},y_{1} ight),\ \left( x_{2},y_{2} ight),(x_{3},y_{3})$ and $\left( a_{1},b_{1} ight),\ \left( a_{2},b_{2} ight),(b_{3},b_{3})$ must be congruent. Which of the statement given above is/are correct?

Question

Consider the following statements : 1. The determinants $\left| \begin{matrix} 1 & a & bc \ 1 & b & ca \ 1 & c & ab \end{matrix} ight|$ and $\left| \begin{matrix} 1 & a & a^{2} \ 1 & b & b^{2} \ 1 & c & c^{2} \end{matrix} ight|$ are not identically equal. 2. For $a > 0,\ b > 0,\ c > 0$ the value of the determinant $\left| \begin{matrix} a & b & c \ b & c & a \ c & a & b \end{matrix} ight|$ is always positive. 3.If$\ \left| \begin{matrix} x_{1} & y_{1} & 1 \ x_{2} & y_{2} & 1 \ x_{3} & y_{3} & 1 \end{matrix} ight| = \left| \begin{matrix} a_{1} & b_{1} & 1 \ a_{2} & b_{2} & 1 \ a_{3} & b_{3} & 1 \end{matrix} ight|$, then the two triangles with vertices $\left( x_{1},y_{1} ight),\ \left( x_{2},y_{2} ight),(x_{3},y_{3})$ and $\left( a_{1},b_{1} ight),\ \left( a_{2},b_{2} ight),(b_{3},b_{3})$ must be congruent. Which of the statement given above is/are correct?

Dr. Abubacker · Accepted Answer

(d)
All statements are false.

Explanation

Reviewed by Dr. Abubacker

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