Week: 1 Mathematics 2

Publish Date: June 10, 2024

Determinants Basics

Understanding Determinants with a 2x2 Matrix

A determinant is a scalar value that can be computed from the elements of a square matrix and encodes certain properties of the matrix. Let’s consider a 2x2 matrix:

To find the determinant of , denoted as or , we use the following formula:

Example:

Consider the matrix:

Step 1: Assign values to , , , and :

Step 2: Calculate the determinant using the formula:

So, the determinant of matrix is .

Understanding Determinants with a 3x3 Matrix

A determinant is a scalar value that can be computed from the elements of a square matrix and encodes certain properties of the matrix. Let’s consider a 3x3 matrix:

To find the determinant of , denoted as or , we use the following formula:

Example:

Consider the matrix:

Step 1: Assign values to , , , , , , , , and :

Step 2: Calculate the determinant using the formula:

So, the determinant of matrix is .

Interpretation:

If the determinant is zero, then the matrix is singular, meaning it does not have an inverse.
If is non-zero, then the matrix is non-singular and has an inverse.
The sign of indicates whether the transformation represented by the matrix preserves orientation (positive) or reverses it (negative) in geometric terms.

Interpretation:

If the determinant is zero, then the matrix is singular, meaning it does not have an inverse.
If is non-zero, then the matrix is non-singular and has an inverse.
The sign of indicates whether the transformation represented by the matrix preserves orientation (positive) or reverses it (negative) in geometric terms.