Problem 73 - Algebra and Trigonometry

Question

What happens to the value of a second-order determinant if the two columns are interchanged?

Step-by-Step Solution

Answer

When the two columns of a second-order determinant are swapped, the value of the determinant is negated.

Step 1: Understand Second Order Determinant

A second order determinant is written in the format: \[ \Delta = \begin{vmatrix} a & b \\ c & d \end{vmatrix}\], and its value is computed as \(ad - bc\).

Step 2: Swapping Columns

If the two columns of this determinant are swapped, then the determinant changes to \[ \Delta = \begin{vmatrix} b & a \\ d & c \end{vmatrix}\]. The value of this new determinant is \(bc - ad\).

Step 3: Compare Values

The value of the new determinant after swapping columns is negative of original second order determinent i.e., \(- \Delta\)