Discuss the properties of determinants. Use examples to elaborate your response.

The determinant is a scalar value that can be used to represent various properties of a square matrix, such as its invertibility and the volume of the parallelogram spanned by its columns or rows. Determinants are widely used in linear algebra, and they play a crucial role in many applications, such as solving systems of linear equations, computing eigenvalues and eigenvectors, and finding the inverse of a matrix.

In this response, we will discuss some of the important properties of determinants that are relevant to linear algebra and its applications.

1. Linearity Property One of the most important properties of determinants is their linearity. Given two matrices A and B and two scalars α and β, the determinant of the matrix αA + βB is equal to α times the determinant of A plus β times the determinant of B, i.e., |αA + βB| = α|A| + β|B|. This property allows us to calculate the determinant of a matrix that is the sum of two or more matrices in _____ _____ __________ ____ ________ ________ ___ _______ ___ ____ ___.
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