Elementary matrix transformations Guide, Meaning , Facts, Information and Description

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:

1. Row switching transformations,
2. Row multiplying transformations,
3. Linear combinator transformations.

1. Row switching transformations

• The matrix T_ij is square.

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

Since the determinant of the identity matrix is unity, det[T_ij]=-1. It follows that for any conformable square matrix A: det[T_ijA]=-det[A].

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

The matrix and its inverse are diagonal matrices.

det[T_i(m)]=m. Therefore for a conformable square matrix A: det[T_i(m)A]=m det[A].

• T_ij(m)^-1=T_ij(-m) (inverse matrix).

The matrix and its inverse are triangular matrices.

det[T_ij(m)]=1. Therefore, for a conformable square matrix A: det[T_ij(m)A]=det[A].

• Linear algebra
• Gauss elimination method

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