MATH 323 Lecture 6

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Elementary Matrices

System of equations can be converted to , where and . is an elementary matrix of type 1, 2, or 3.


  • Premultiplication performs rule operations on rows
  • Postmultiplication performs rule operations on columns
  • is an elementary matrix of the same type.

Row Equivalence

describes that a matrix can be converted to another matrix by applying elementary matrices:

If is nonsingular, then is row equivalent to :

Therefore, the augmented matrix can be converted to by performing elementary row operations to convert it to reduced row echelon form.


Therefore, if we have the system of equations , the solution is .

Triangular Matrices

Suppose is a square matrix.

  • If is of the form , it is diagonal.
  • If is of the form , it is upper triangular.
  • If is of the form , it is lower triangular.

Triangular Factorization

If a matrix can be reduced to strict upper triangular form using row operations of type 3, then , where – unit lower triangular matrix and strictly upper triangular matrix.

Each operation is of the form , where and are row indices of .

, , and .

If we construct a matrix by replacing elements of with corresponding values, we get such that and , where is a single replacement in for each step.


For a square matrix , there is a corresponding scalar determinant of

Written in any of the following formats:

A matrix is singular iff its determinant is 0.

  1. For a square matrix,
  2. For a square matrix, (product of main diagonal minus product of "secondary diagonal")
  3. For a square matrix, (See MATH 251 Lecture 10 for hints and tricks)