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Types of Matrices

Certain types of matrices, which play important roles in matrix theory, are now considered.

Row Matrix. A matrix that has exactly one row is called a row matrix. For example, the matrix

A = [5   2  -1   4]

is a row matrix of order 1 x 4

Column Matrix. A matrix consisting of a single column is called a column matrix. For example, the matrix

B =

is a 3 x 1 column matrix

Zero or Null Matrix. An m x n matrix whose entries are all 0 is called the m x n zero or null matrix. It is usually denoted by Om x n or more simply by O. For example,

O =

is a 2 x 4 zero matrix

Square Matrix. An m x n matrix is said to be a square matrix of order n if m = n. That is, if it has same number of columns as rows.

For example,



are square matrices of order 3 and 2 respectively

In a square matrix A = aij of order n, the entries a11, a22,.....,ann which lie on the diagonal extending from the left upper corner to the lower right corner are called the main diagonal entries, or more simply the main diagonal. Thus in the matrix



the entries a11 = 3, a22 = 6 and a33 = 8 constitute the main diagonal.

Triangular Matrix. A square matrix is said to be an upper (lower) triangular matrix if all entries below the main diagonal are zeros.

For example,



are upper and lower triangular matrices, respectively.

Diagonal Matrix. A square matrix is said to be diagonal if each of its entries not falling on the main diagonal is zero. Thus a square matrix A = (aij ) is diagonal if ( aij) = 0 I j. For example.



is a diagonal matrix.

Notation. A diagonal matrix A of order n with diagonal elements a11, a22, ann is denoted by


Scalar Matrix. A diagonal matrix whose all the diagonal elements are equal is called a scalar matrix. For example,



is a scalar matrix

Identity Matrix or Unit Matrix. A square matrix is said to be identity matrix or unit matrix if all its main diagonal entries are 1’s and all other entries are 0’s. An identity matrix of order n is denoted by In or more simply by I. For example,

 I3


is identity matrix of order 3.

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