An matrix with entries from a field is a rectangular array of the form

Where all the entries are elements of .
A matrix is square if it has the same number of rows and columns
Two matrices are equal if and only if
The set of matrices is a Vector Space denoted under matrix (element-wise) addition and scalar multiplication.
The trace of an matrix, denoted is the sum of all entries of in the diagonal. That is
- (Friedberg e1.2.6) The set of matrices having trace equal to is a subspace of
Every matrix is equivalent to a linear transformation
- The rank of is defined as
- An matrix is invertible if and only if its rank is
- (Friedberg 3.5) The rank of any matrix equals the maximum number of its linearly independent columns.
  
  That is, the rank is the dimension of the subspace generated by the columns.
- (Friedberg 3.6.2) Let )
  - The rank of any matrix equals the maximum number of its linearly independent row (i.e., it is the dimension of the subspace generated by its rows.
  - The rows and columns of any matrix generate subspaces of the same dimension
- (Friedberg e3.2.8) For any ,

Special Matrices

The transpose of an matrix is the matrix where
A matrix is symmetric if
- The set of symmetric matrices is a subspace of
- The transpose can be seen in the context of dual spaces
A matrix is skew-symmetric if
- (Friedberg e1.3.26) The set of skew-symmetric matrices is a subspace of . In fact, let be the set of skew-symmetric and the set of symmetric matrices. Then
  
  Every matrix is the sum of a symmetric and skew symmetric matrix.
A matrix is diagonal if whenever .
- The set of diagonal matrices is a subspace of
A Vandermonde Matrix is a matrix of the form
- (Friedberg e4.3.12c) The determinant of the Vandermonde matrix is given by
A matrix is nilpotent if for some positive

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