# Write the value of the following determinant : $\begin {vmatrix} a-b & b-c & c-a \\ b-c & c-a & a-b \\ c-a & a-b & b-c \end {vmatrix}$

Toolbox:
• (i) If any two columns or rows of a determinant are identical(all corresponding elements are same),then the value of the determinant is zero.
• (ii) If some or all elements of a row or column of a determinant can be expressed as sum of two (or more) determinants.
Step 1:
By using the property of determinants let us split the determinant as
$\begin{vmatrix}a & b& c\\b & c & a\\c & a & b\end{vmatrix}-\begin{vmatrix}b & c & a\\c & a & b\\a & b & c\end{vmatrix}$
Step 2:
The second determinant can be rearranged as
$\begin{vmatrix}a & b & c\\b & c & a\\c & a & b\end{vmatrix}$
Step 3:
Hence $\begin{vmatrix}a-b & b-c & c-a\\b-c & c-a & a-b\\c-a & a-b & b-c\end{vmatrix}=\begin{vmatrix}a & b & c\\b & c & a\\c & a & b\end{vmatrix}-\begin{vmatrix}a & b & c\\b & c & a\\c & a & b\end{vmatrix}$=0.
answered Apr 9, 2013