MATH 323 Lecture 5

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Elementary Matrices

System of equations now in form , where is , and and are

For a non-singular, invertible matrix , , and this system will be equivalent to the original. could even be product of several matrices: , where is an elementary matrix.

Types of elementary matrices (each corresponds to rules of matrix):

  1. obtained by interchanging two rows of identity matrix
    • permutes rows
    • permutes columns
  2. obtained by multiplying row of identity matrix by a nonzero constant
    • scales corresponding row by
    • scales corresponding column by .
  3. obtained from by adding a multiple of one row to another row :
    • adds multiple of row to row
    • adds multiple of column to column
Note: (and thus ) is always a square matrix, but can be of any size

is a matrix: we can think of it as being obtained from by either a row operation or column operation. If is a matrix, premultiplying by has the effect of performing the same row operation on . If is a matrix, postmultiplying by is equivalent to performing that same column operation on .

Theorem 1.4.1

If is an elementary matrix, then is nonsingular and is an elementary matrix of the same type as

  1. , since interchanging two rows twice undoes the effect, so
  2. if has an at position . will have at
  3. has a at position , where . will have at . In multiplying by , position will be product of th row and th column:

Row Equivalence

is row equivalent to (written iff there exists a finite sequence of elementary matrices such that . In other words, can be obtained from by a finite number of row operations.

By extension, two augmented matrices and are row equivalent iff and are equivalent systems.

Theorem 1.4.2

Equivlent Conditions for Nonsingularity

Let be a matrix. Then the following are equivalent:

  1. is nonsingular
  2. has only trivial solution
  3. is row equivalent to identity matrix of size ()

(1) → (2): Let be solution of . So can become . , so

(2) → (3): Rewrite in row echelon form as . If one diagonal entry of is 0, then there is at least one free variable and infinitely many solutions, one of which is nonzero. This leads to a contradiction, so all diagonal entries of must be 1, so rref will be identity matrix.

(3) → (1): If is row equivalent to , then there exists a sequence of elementary matrices such that . Therefore, (note reverse order), so is nonsingular.

Corrolary 1.4.3

The system of equations has a unique solution iff is nonsingular.

  • is nonsingular, can be rewritten as
  • Assume unique solution exists. If were singular then homogeneous system would have a nonzero solution (by Theorem 1.4.2). If this were the case, ... there would be another solution, which is a contradiction.