Determinants

4.1 Introduction

This section explains how determinants relate to matrices and linear systems. A unique solution for a system of linear equations (like ( ax + by = c )) is determined by the determinant of the coefficient matrix. The determinant indicates whether a system has a unique solution, with applications spanning various fields including engineering and economics. We focus on determinants of order three or less.

4.2 Determinant

A determinant is a scalar value that can be associated with a square matrix. For a matrix ( A = [a_{ij}] ) of order n, the determinant is denoted as ( \text{det}(A) ), |A|, or ( \Delta ). Only square matrices (n x n) can have a determinant:

Notation:

• det or |A| to denote the determinant.

Example of determinants for matrices of different orders:

1. Order 1: For ( A = [a] ), ( \text{det}(A) = a )
2. Order 2: For a 2x2 matrix ( A = \begin{pmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{pmatrix} ), ( \text{det}(A) = a_{11}a_{22} - a_{12}a_{21} )
3. Order 3: A determinant can be computed by expanding along any row or column.

4.2.3 Determinant of a 3x3 Matrix

To compute the determinant of a 3x3 matrix:

• We can expand along a row or column.
• Example expansion along the first row: [ |A| = a_{11}C_{11} + a_{12}C_{12} + a_{13}C_{13} ] Where ( C_{ij} ) denotes cofactors.

4.3 Area of a Triangle

The area of a triangle formed by points ((x_1, y_1), (x_2, y_2), (x_3, y_3)) can be expressed using determinants. [ \text{Area} = \frac{1}{2} \text{det} \begin{pmatrix} x_1 & y_1 & 1 \ x_2 & y_2 & 1 \ x_3 & y_3 & 1 \end{pmatrix} ]

4.4 Minors and Cofactors

• Minor: Denoted as ( M_{ij} ), it's the determinant formed by deleting row i and column j.
• Cofactor: Denoted as ( A_{ij} = (-1)^{i+j} M_{ij} )
• Determinants can also be represented as sum of elements in a row (or column) multiplied by their corresponding cofactors.

4.5 Adjoint and Inverse of a Matrix

• Adjoint of matrix ( A ): The transpose of the cofactor matrix.
• A matrix is singular if its determinant is zero, and non-singular otherwise.
• For a matrix ( A ), ( A^{-1} = \frac{\text{adj}(A)}{|A|} ) if ( |A| \neq 0 ).

4.6 Applications of Determinants and Matrices

Determinants can help solve systems of linear equations:

• A consistent system has at least one solution.
• Analysis of determinants leads to unique solutions or indicates inconsistency.

We use the matrix form ( AX = B ) for representing systems of equations, where ( X ) contains variables, and employing determinants helps determine the presence and uniqueness of solutions.

1. The determinant indicates the nature of solutions for systems of linear equations.
2. Determinants only exist for square matrices.
3. For a 2x2 matrix, ( |A| = a_{11}a_{22} - a_{12}a_{21} ).
4. For a 3x3 matrix, determinants can be expanded along any row or column.
5. The area of a triangle can be calculated using determinants.
6. Minors and cofactors are foundational in calculating determinants.
7. The adjoint of matrix A is used to find the inverse of A.
8. A matrix is singular if ( |A| = 0 ), indicating no unique inverse.
9. Determinants serve to analyze the consistency of linear systems of equations.
10. Use the formula ( X = A^{-1}B ) for matrix methods to solve linear equations.