Problem 73 - College Algebra

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 interchanged, the value of the determinant is multiplied by -1.

Step 1: Determine the determinant of the first order

Let's consider a second-order determinant with elements \(a, b, c, d\). The expression for the determinant is \(By = ad - bc\)

Step 2: Interchange the columns

When the columns of the matrix are switched, the positions of \(b\) and \(c\) are swapped, and the new matrix has elements \(b, a, d, c\)

Step 3: Determine the determinant of the new order

Now calculate the new determinant \(By' = ac - bd\)

Step 4: Compare the determinants

By comparing the determinants \(By\) and \(By'\), it is observed that we obtain the negative of the original determinant when we switch the two columns of a second-order determinant. This happens because the process of switching columns is equivalent to multiplying the determinant by -1