Problem 65 - 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 changes its sign.

Step 1: Understanding Determinants

In a second order determinant \( \begin{bmatrix} a & b \\ c & d \end{bmatrix} \), the determinant can be computed as \( ad - bc \). It is important to note that interchanging columns or rows simply changes the sign of the determinant.

Step 2: Interchange the Columns

Now, assume we interchange the columns in this determinant, we will have \( \begin{bmatrix} b & a \\ d & c \end{bmatrix} \). The determinant for this will be \( cb - da \).

Step 3: Compare the Two Determinants

On comparing the two determinants, you will observe that \( cb - da = -(ad - bc) \). The determinants are equal in absolute value but the signs of the values are opposite. Hence, when the two columns of a second order determinant are interchanged, the value of the determinant changes its sign.