MATHEMATICS S5 UNIT 7: Matrices and Determinant of Order 3.

Matrices and determinants of order 3 are fundamental concepts in linear algebra, playing crucial roles in various mathematical and scientific applications.

I. Matrices of Order 3 (3×3 Matrices)

A matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. The “order” of a matrix specifies its dimensions (number of rows $\times$ number of columns).

A matrix of order 3 specifically refers to a square matrix that has 3 rows and 3 columns.

General Form:  A 3×3 matrix, let’s call it $A$, is typically represented as:

Where aij​ represents the element located in the $i$-th row and $j$-th column.

Key Concepts Related to 3×3 Matrices:

• Elements: The individual numbers or expressions within the matrix.
• Rows: The horizontal lines of elements.
• Columns: The vertical lines of elements.
• Main Diagonal: The elements from the top-left to the bottom-right (a11,a22,a33​).
• Identity Matrix (I3​): A special 3×3 matrix with 1s on the main diagonal and 0s elsewhere. It acts like the number ‘1’ in matrix multiplication:

• Zero Matrix: A matrix where all elements are zero.
• Matrix Operations: You can perform operations like addition, subtraction, scalar multiplication, and matrix multiplication with 3×3 matrices, following specific rules.

Uses of 3×3 Matrices:

• Representing Systems of Linear Equations: A system of three linear equations with three variables can be compactly written as a matrix equation $AX=B$.
• Linear Transformations in 3D Space: Matrices can represent geometric transformations like rotations, scaling, reflections, and shears in three-dimensional space.
• Data Organization: Used in various fields (e.g., computer graphics, physics, engineering, economics) to organize and manipulate data.

II. Determinant of Order 3.

The determinant is a special scalar (a single number) value that can be computed from the elements of a square matrix. It provides crucial information about the matrix, particularly its invertibility and the geometric scaling effect of the linear transformation it represents.

The determinant of a matrix $A$ is denoted as $\det \left(A\right)$ or $|A|$.

Significance of the Determinant:

• Invertibility: A square matrix $A$ is invertible (meaning its inverse A⁻¹ exists) if and only if its determinant $\det \left(A\right)=0$. If $\det \left(A\right)=0$, the matrix is called singular, and it does not have an inverse.
• Systems of Linear Equations: For a system $AX=B$, if $\det (A)=0$, there is a unique solution. If $\det \left(A\right)=0$, there might be no solution or infinitely many solutions.
• Geometric Interpretation:
  • In 3D space, the absolute value of the determinant of a 3×3 matrix represents the volume scaling factor of the linear transformation represented by the matrix. For example, if a transformation represented by matrix $A$ is applied to a unit cube, the volume of the transformed shape will be $|\det (A)|$.
  • The sign of the determinant indicates whether the transformation preserves or reverses the orientation of the space.

Methods for Calculating the Determinant of a 3×3 Matrix:

There are two primary methods for calculating the determinant of a 3×3 matrix:

1. Cofactor Expansion (Laplace Expansion): This method is generalizable to matrices of any size. To compute the determinant using cofactor expansion, you pick any row or any column. For each element in that chosen row/column, you multiply the element by its cofactor and then sum these products.

Minor (Mij​): The determinant of the submatrix formed by deleting the $i$-th row and $j$-th column of the original matrix. For a 3×3, minors will be determinants of 2×2 matrices.