Problem-10

determinants

Question

Compute \(\begin{vmatrix} 1 & a & bc\\ 1 & b & ca\\ 1 & c & ab \end{vmatrix}\).

Hint

Row operations:

  • \(R_2 \rightarrow R_2 - R_1\)
  • \(R_3 \rightarrow R_3 - R_1\)

Solution

There are four steps here, each one corresponding to one row operation. The sequence of steps is as follows:

  1. \(R_2 \rightarrow R_2 - R_1\)
  2. \(R_3 \rightarrow R_3 - R_1\)
  3. Take \((b - a)\) and \((c - a)\) out of row-2 and row-3 respectively. This is the scaling operation, but performed slightly differently.
  4. Expand the determinant along the first column.

This is how it works out:

\[ \begin{equation*} \begin{aligned} \begin{vmatrix} 1 & a & bc\\ 1 & b & ca\\ 1 & c & ab \end{vmatrix} & =\begin{vmatrix} 1 & a & bc\\ 0 & b-a & c( a-b)\\ 0 & c-a & b( a-c) \end{vmatrix}\\ & \\ & =( b-a)( c-a)\begin{vmatrix} 1 & a & bc\\ 0 & 1 & -c\\ 0 & 1 & -b \end{vmatrix}\\ & \\ & =( b-a)( c-a)( c-b)\\ & \\ & =( a-b)( b-c)( c-a) \end{aligned} \end{equation*} \]