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Chapter 6: Problem 65

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

Short Answer

Expert verified
When the two columns of a second order determinant are interchanged, the value of the determinant changes its sign.

Step by step solution

01

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.
02

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 \( cb - da \).
03

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.

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Most popular questions from this chapter

Without going into too much detail, describe how to solve a linear system in three variables using Cramer's rule.

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