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Types of matrices

Definition.
A square matrix is a matrix in which the number of rows equals the number of columns (size n×n), the number n, is called the order of the matrix.
Example.
( 4  1  -7 ) - a square matrix with size 3×3
 -1  0  2 
 4  6  7 

Definition.
The zero matrix is a matrix whose entries are equal to zero, ie, aij = 0, ∀i, j.
Example.
( 0  0  0 ) - zero matrix
 0  0  0 

Definition.
Row vector is a matrix consisting of a one row.
Example.
( 1  4  -5 ) - row vector

Definition.
Column vector is a matrix consisting of a one column.
Example.
( 8 ) - column vector
 -7 
 3 

Definition.
Diagonal matrix is a square matrix whose entries standing outside the main diagonal are equal zero.
Example of diagonal matrix.
( 4  0  0 ) - diagonal entries are arbitrarynot diagonal entries are equal to zero
 0  5  0 
 0  0  0 

Definition.
Identity matrix is a diagonal matrix in which all the elements on the main diagonal are equal to 1.
Denote.
The identity matrix usually denoted by I.
Example of identity matrix.
I( 1  0  0 ) - diagonal entries are equal to 1not diagonal entries are equal to 0
 0  1  0 
 0  0  1 

Definition.
Upper triangular matrix is a matrix whose elements below the main diagonal are equal to zero.
Example of upper triangular matrix.
( 7  -6  0 )
 0  1  6 
 0  0  0 

Definition.
Lower triangular matrix is a matrix whose elements above the main diagonal are equal to zero.
Example of lower triangular matrix.
( 7  0  0 )
 6  1  0 
 -2  0  5 

N.B. The diagonal matrix is a matrix that is both upper triangular and lower triangular.


Definition.
Specifically, a matrix is in row echelon form if:
  • all nonzero rows are above all zero rows;
  • if the first non-zero element of a row is located in a column with the number i, and the next line is not zero, then the first non-zero element of the next line should be in the column with number greater than i.
Examples of row echelon form of matrix.
( 7  0  8 )
 0  0  4 
( 7  0  8  8  8 )
 0  0  1  3  5 
 0  0  0  -3  5 
 0  0  0  0  0 
 0  0  0  0  0 

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