Question

What happens to the determinant of a matrix if you switch two columns?

a) It stays the same
b) It becomes zero
c) It becomes negative
d) It changes sign

Answer 1

Switching two columns in a matrix causes the determinant to change sign. If the original determinant was positive, it becomes negative, and vice-versa.

Step-by-step explanation:

When you switch two columns in a matrix, the determinant of the matrix changes sign. This means that if the determinant was positive before the switch, it will become negative, and if it was negative, it will become positive.

Here's a basic step-by-step explanation to understand why this happens:

Consider a square matrix and calculate its determinant.

Switch any two columns of the matrix.

Calculate the determinant of the modified matrix.

You will notice that the determinant of the matrix after switching two columns is the negative of the original determinant. Thus, the correct answer to the question 'What happens to the determinant of a matrix if you switch two columns?' is that it changes sign.