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Questions  >>  CBSE XII  >>  Math  >>  Determinants

If a,b,c are positive and unequal,show that the value of the determinant $\Delta=\begin{vmatrix}a & b & c\\b & c & a\\c & a & b\end{vmatrix}$ is negative.

1 Answer

  • Elementary transformation can be made by
  • (i) In terchanging the rows or columns.
  • (ii) The addition to the elements of any rows or columns.
  • If $A=\begin{bmatrix}a_{11} & a_{12} & a_{13}\\a_{21} & a_{22} & a_{23}\\a_{31} & a_{32} & a_{33}\end{bmatrix}$
  • $|A|=a_{11}(a_{22}\times a_{33}-a_{23}\times a_{32})-a_{12}(a_{21}\times a_{33}-a_{23}\times a_{31})+a_{13}(a_{21}\times a_{32}-a_{22}\times a_{31})$
Step 1:Let $\Delta=\begin{vmatrix}a & b& c\\b & c & a\\c & a & b\end{vmatrix}$
Apply $C_1\rightarrow C_1+C_2+C_3$
Hence $\Delta=\begin{vmatrix}a+b+c & b & c\\a+b+c & c & a\\a+b+c & a & b\end{vmatrix}$
Take (a+b+c) as the common factor from $C_1$
Therefore $\Delta=\begin{vmatrix}1 & b& c\\1 & c & a\\1 & a & b\end{vmatrix}$
Apply $R_2\rightarrow R_2-R_1$ and $R_3\rightarrow R_3-R_1$
$\Delta=(a+b+c)\begin{vmatrix}1 & b & c\\0 & c-b & a-c\\0 & a-b & b-c\end{vmatrix}$
Step 2:Now let us expand along $C_1$
Now multiply and divide by -2
But we know$(a+b)^2=a^2+2ab+b^2$
This is negative,since a+b+c>0 and $(a-b)^2+(b-c)^2+(c-a)^2>0.$
Hence proved.
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