Publish Date: June 10, 2024

Determinants Properties

1. Determinant of Matrix

The determinant of a matrix is a scalar value that can be calculated using various methods such as cofactor expansion or Laplace’s formula.

Consider the matrix :

The determinant of both matrices are:

As you can see, both determinants are equal.

2. Interchange of Columns

If we want to interchange the first and second columns of matrix , the resulting matrix would be:

Original matrix:
After interchanging columns:

The determinants of both matrices are:

Both determinants remain the same after interchanging columns.

4. Determinant of Transpose

Let’s delve into the concept of the determinant of a transpose using a 3x3 matrix as an example.

Consider the matrix :

The transpose of , denoted as , is obtained by swapping the rows and columns of , resulting in:

Now, the determinant of , denoted as , is a scalar value that can be calculated using various methods such as cofactor expansion or Laplace’s formula.

Similarly, the determinant of , denoted as , is the scalar value obtained by applying the same determinant calculation method to the transposed matrix.

Let’s find the determinant of :

Now, let’s find the determinant of :

As we can see, the determinants of and are equal.

5. Determinant of Product

The determinant of a product of two matrices is the product of their determinants.

Let’s find the determinant of the product of and , where :

Now, let’s find the determinant of :

Let’s calculate the determinants of and :

Now, let’s find the product of the determinants:

As we can see, the determinant of the product of and is indeed equal to the product of their determinants.

6. Inverse of Determinant

The determinant of the inverse of a square matrix has a straightforward relationship with the determinant of . If is an invertible matrix, meaning its determinant is non-zero, then:

In other words, the determinant of the inverse of a matrix is the reciprocal of the determinant of the original matrix.

Let’s illustrate this with an example:

Suppose we have a matrix:

First, let’s find the determinant of :

Since is invertible (because its determinant is non-zero), let’s find its inverse, :

Now, let’s find the determinant of :

As expected, .