Properties of Determinant

In this article, I have discussed some of the properties of determinant. Determinants are mathematical objects that are associated with square matrices. They have several properties that make them useful in various mathematical applications. Here are some key properties of determinants:

Dependence on Matrix Size:

Determinants are defined only for square matrices, i.e., matrices with an equal number of rows and columns.

Scalar Multiplication:

If a matrix has a scalar multiple, multiplying a row or column of the matrix by a scalar multiplies the determinant by the same scalar.

Row or Column Interchange:

Interchanging any two rows or columns of a matrix changes the sign of the determinant.

Linearity in Rows or Columns:

The determinant is a linear function of each row (or column) when the other rows (columns) are held constant.

Row or Column Addition:

If a matrix has a row (or column) that can be expressed as the sum of two other rows (columns), the determinant of the matrix remains unchanged if that row (column) is replaced by the sum of the other two rows (columns).

Determinant of the Identity Matrix:

The determinant of the identity matrix is 1.

Product of Matrices:

The determinant of a product of two matrices is the product of their determinants: det(AB) = det(A) * det(B).

Inverse of a Matrix:

The determinant of the inverse of a matrix is the reciprocal of the determinant of the original matrix: det(A^(-1)) = 1 / det(A), provided A is invertible.

Transpose of a Matrix:

The determinant of a matrix and its transpose are equal: det(A^T) = det(A).

Singular Matrices:

If a matrix is singular (non-invertible), i.e., its determinant is zero, then the system of linear equations represented by the matrix has no unique solution.

Triangle Matrices:

The determinant of an upper (lower) triangular matrix is the product of its diagonal elements.

These properties make determinants a powerful tool in linear algebra, used in various applications such as solving systems of linear equations, computing matrix inverses, and studying linear transformations.