Matrix

Rank of a matrix

Full rank matrix

A matrix is of full rank if its rank is equal to the smallest of the number of its rows or columns.


A full-rank square matrix is invertible.

Used to solve systems of linear equations

Substitution

Elimination

Matrix methods

Row reduction

Gaussian elimination

Row echelon form

Row reduced echelon form

Convert a matrix to Row Echelon Form using elementary row operations:


  1. Forward Elimination
  2. Back Substitution

Rules

  1. All non-zero rows are above any rows of all zeros.
  2. Leading coefficient of a non-zero row is to the right of the leading coefficient of the row above it.

Rules

  1. Same as Row Echelon Form, plus:
  2. Leading coefficient of each non-zero row is 1 and is the only non-zero entry in its column.

Trace of a matrix

The maximum number of linearly independent row or column vectors in the matrix.


Importance:

  1. Determines the dimension of the column space.
  2. Related to the solutions of linear equations.

The sum of the elements on the main diagonal of a square matrix.


Properties:

  1. Invariant under matrix similarity.
  2. Used in various matrix theorems.

Inverse of a matrix

A matrix that, when multiplied by the original matrix, yields the identity matrix.
Conditions:

  1. Only square matrices have inverses.
  2. A matrix has an inverse if and only if it is of full rank.