Elementary Matrices in Linear Algebra: Identity Matrices and Elementary Row Operations

Learn about elementary matrices, including identity matrices and how they relate to elementary row operations. This guide provides definitions, examples, and explanations of their properties and use in linear algebra.

Elementary Matrices in Discrete Mathematics

What is a Matrix?

A matrix is simply a grid of numbers arranged in rows and columns. These numbers are called the matrix's elements or entries. A matrix with 'n' rows and 'm' columns is called an n x m matrix. A square matrix has an equal number of rows and columns (n x n).

Example: A 3 x 3 matrix:

A = [a11, a12, a13]
    [a21, a22, a23]
    [a31, a32, a33]
        

What is an Identity Matrix?

An identity matrix is a special square matrix. It has 1s along its main diagonal (top-left to bottom-right) and 0s everywhere else. It's always a square matrix (n x n).

Examples:

I2 = [1, 0]
          [0, 1]

I3 = [1, 0, 0]
          [0, 1, 0]
          [0, 0, 1]
        

What is an Elementary Matrix?

An elementary matrix is a square matrix created by performing a single, simple operation on an identity matrix. These operations can be applied to either rows or columns:

Row Operations:

• R1: Multiply all elements in a row by a non-zero number.
• R2: Swap two rows.
• R3: Add a multiple of one row to another row.

Column Operations: (Similar operations apply to columns)

• C1: Multiply all elements in a column by a non-zero number.
• C2: Swap two columns.
• C3: Add a multiple of one column to another column.

Examples of Elementary Matrices

1. Multiply the second row of I₂ by -3:

E = [1, 0]
    [0, -3]
        

2. Swap the second and fourth rows of I₄:

E = [1, 0, 0, 0]
    [0, 0, 1, 0]
    [0, 1, 0, 0]
    [0, 0, 0, 1]
        

Properties of Elementary Matrices

• They are always square.
• They are invertible (have an inverse).
• Multiplying a matrix on the left by an elementary matrix performs a row operation; multiplying on the right performs a column operation.
• They are crucial in Gaussian and Gauss-Jordan elimination methods.

Inverses of Elementary Matrices

Each row operation has a corresponding inverse operation:

• Inverse of R_i → kR_i is R_i → (1/k)R_i
• Inverse of R_i → kR_j + R_i is R_i → -kR_j + R_i
• Swapping rows is its own inverse.

Example of Elementary Row Operation

Let's say we have matrix A:

A = [1, 2, 3]
    [4, 5, 6]
    [7, 8, 9]
        

Multiplying the first row by 2 (an elementary row operation):

E * A = [2, 4, 6]
        [4, 5, 6]
        [7, 8, 9]
        

Properties of Invertible Matrices and Elementary Matrices

If a matrix A is invertible, it can be written as a product of elementary matrices. This fact has several consequences regarding the properties of invertible matrices.

• A matrix that swaps two rows is its own inverse.
• If an elementary matrix multiplies a row by 'm', its inverse divides that row by 'm'.
• If an elementary matrix adds 'm' times a row to another row, its inverse subtracts 'm' times that row.

Examples of Elementary Matrix Verification

Example 1: Is matrix A elementary?

Solution: No, because an elementary matrix must be square, and A (if it's not square) is not.

Example 2: Is matrix A elementary?

Solution: Yes, if A is obtained from an identity matrix by a single row or column operation.

char ch = 'a';
            

She said "Hello!" to me.