Question-8

Find out the determinant of the following matrix:

\[ \begin{bmatrix}1 & a & bc\\ 1 & b & ca\\ 1 & c & ab \end{bmatrix} \]

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\(R_{2}\rightarrow R_{2}-R_{1}\)

\[ \begin{bmatrix}1 & a & bc\\ 0 & b-a & c(a-b)\\ 1 & c & ab \end{bmatrix} \]

\(R_{3}\rightarrow R_{3}-R_{1}\)

\[ \begin{bmatrix}1 & a & bc\\ 0 & b-a & c(a-b)\\ 0 & c-a & b(a-c) \end{bmatrix} \]

Both these operations leave the determinant unchanged. We can now expand the determinant along the first column:

\[ \begin{aligned} \begin{vmatrix}1 & a & bc\\ 1 & b & ca\\ 1 & c & ab \end{vmatrix} & =b(b-a)(a-c)-c(c-a)(a-b)\\ \\ & =(a-b)(c-a)(b-c)\\ \\ & =(a-b)(b-c)(c-a) \end{aligned} \]