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