Elementary matrix transformations

Elementary matrix transformations or Elementary row and column transformations are linear transformations which are normally used in gauss elimination to solve a set of linear equations.

We distinguish three types of elementary transformations and their corresponding matrices:

Row switching transformations,
Row multiplying transformations,
Linear combinator transformations.

Table of contents

1 1. Row switching transformations

1.1 Properties

2 2. Row multiplying transformations

2.2 Properties

3 3. Linear combinator transformations

3.3 Properties

1. Row switching transformations

This transformation, T_ij, switches all matrix elements on row i with their counterparts on row j. The matrix resulting in this transformation is:

Properties

The inverse of this matrix is itself: T_ij^-1=T_ij.

When applied to a matrix A: det[TA]=-det[A].

The matrix and it's inverse are lower triangular matrices.

2. Row multiplying transformations

This transformation, T_i(m), multiplies all elements on row i with m. The matrix resulting in this transformation is:

Properties

The inverse of this matrix is: T_i(m)^-1=T_i(1/m).

When applied to a matrix A: det[TA]=mdet[A].

The matrix and it's inverse are lower triangular matrices.

3. Linear combinator transformations

This transformation, T_ij(m), substracts row i multiplied by m from row j. The matrix resulting in this transformation is:

Properties

The inverse of this matrix is: T_ij(m)^-1=T_ij(-m).

When applied to a matrix A: det[TA]=det[A].

The matrix and it's inverse are lower triangular matrices.