Problem-11

Question

Compute \(\displaystyle \begin{vmatrix} a & b & c\\ b & c & a\\ c & a & b \end{vmatrix}\).

Hint

Row operations:

• \(R_1 \rightarrow R_1 + R_2 + R_3\)
• \(C_1 \rightarrow C_1 - C_2\)
• \(C_2 \rightarrow C_2 - C_3\)

Solution

\[ \begin{aligned} \begin{vmatrix} a & b & c\\ b & c & a\\ c & a & b \end{vmatrix} & =\begin{vmatrix} a+b+c & a+b+c & a+b+c\\ b & c & a\\ c & a & b \end{vmatrix}\\\\ & =( a+b+c) \cdot \begin{vmatrix} 1 & 1 & 1\\ b & c & a\\ c & a & b \end{vmatrix}\\\\ & =( a+b+c) \cdot \begin{vmatrix} 0 & 0 & 1\\ b-c & c-a & a\\ c-a & a-b & b \end{vmatrix}\\\\ & =( a+b+c)\left[( b-c)( a-b) -( c-a)^{2}\right]\\\\ & =( a+b+c)\left[ ab-b^{2} -ac+bc-c^{2} -a^{2} +2ac\right]\\\\ & =( a+b+c)\left[ ab+bc+ca-\left( a^{2} +b^{2} +c^{2}\right)\right] \end{aligned} \]