Elementary Matrices in Linear Algebra

Study Notes

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

There are three types of elementary matrices:

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

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

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

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

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.

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.

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.