Elementary Matrices in Linear Algebra

Definition

An elementary matrix is a square matrix that can be obtained from the identity matrix by performing a single elementary row operation.

Types of Elementary Matrices

There are three types of elementary matrices:

1. Row Swap Matrix

• Obtained by swapping two rows of the identity matrix
• Denoted by Rs(i, j), where i and j are the row indices to be swapped

2. Row Multiplication Matrix

• Obtained by multiplying a row of the identity matrix by a non-zero scalar
• Denoted by Rm(i, k), where i is the row index and k is the scalar

3. Row Addition Matrix

• Obtained by adding a multiple of one row to another row of the identity matrix
• Denoted by Ra(i, j, k), where i and j are the row indices and k is the scalar

Properties

• Elementary matrices are invertible and their inverses are also elementary matrices
• The product of two elementary matrices is also an elementary matrix
• Any invertible matrix can be expressed as a product of elementary matrices

Applications

• Elementary matrices are used to perform row operations on a matrix, which is essential in Gaussian elimination and other linear algebra techniques
• They can be used to find the inverse of a matrix
• They are used in solving systems of linear equations

Elementary Matrices

• An elementary matrix is a square matrix that can be obtained from the identity matrix by performing a single elementary row operation.

Types of Elementary Matrices

• Row Swap Matrix: obtained by swapping two rows of the identity matrix, denoted by Rs(i, j), where i and j are the row indices to be swapped.
• Row Multiplication Matrix: obtained by multiplying a row of the identity matrix by a non-zero scalar, denoted by Rm(i, k), where i is the row index and k is the scalar.
• Row Addition Matrix: obtained by adding a multiple of one row to another row of the identity matrix, denoted by Ra(i, j, k), where i and j are the row indices and k is the scalar.

Properties

• Elementary matrices are invertible and their inverses are also elementary matrices.
• The product of two elementary matrices is also an elementary matrix.
• Any invertible matrix can be expressed as a product of elementary matrices.

Applications

• Elementary matrices are used to perform row operations on a matrix, essential in Gaussian elimination and other linear algebra techniques.
• They can be used to find the inverse of a matrix.
• They are used in solving systems of linear equations.