Define matrix and order of matrix? Explain the various types of matrix
Answers
Answer:
A matrix consists of rows and columns. These rows and columns define the size or dimension of a matrix. The various types of matrices are row matrix, column matrix, null matrix, square matrix, diagonal matrix, upper triangular matrix, lower triangular matrix, symmetric matrix, and antisymmetric matrix
Types of Matrices
Different types of Matrices and their forms are used for solving numerous problems. Some of them are as follows:
1) Row Matrix
A row matrix has only one row but any number of columns. A matrix is said to be a row matrix if it has only one row. In general, A = [aij]1 × n is a row matrix of order 1 × n.
2) Column Matrix
A column matrix has only one column but any number of rows. A matrix is said to be a column matrix if it has only one column.
In general, B = [bij]m × 1 is a column matrix of order m × 1.
3) Square Matrix
A square matrix has the number of columns equal to the number of rows. A matrix in which the number of rows is equal to the number of columns is said to be a square matrix. Thus an m × n matrix is said to be a square matrix if m = n and is known as a square matrix of order ‘n’. In general, A = [aij] m × m is a square matrix of order m.
4) Rectangular Matrix
A matrix is said to be a rectangular matrix if the number of rows is not equal to the number of columns.
5) Diagonal matrix
A square matrix B = [bij] m × m is said to be a diagonal matrix if all its non-diagonal elements are zero, that is a matrix B =[bij]m×m is said to be a diagonal matrix if bij = 0, when i ≠ j.
6) Scalar Matrix
A diagonal matrix is said to be a scalar matrix if all the elements in its principal diagonal are equal to some non-zero constant. A diagonal matrix is said to be a scalar matrix if its diagonal elements are equal, that is, a square matrix B = [bij]n × n is said to be a scalar matrix if
bij = 0, when i ≠ j
bij = k, when i = j, for some constant k.
7) Zero or Null Matrix
A matrix is said to be zero matrix or null matrix if all its elements are zero.
For Example,
A=[0]
are all zero matrices of the order 1, 2 and 3 respectively. We denote zero matrix by O.
8) Unit or Identity Matrix
If a square matrix has all elements 0 and each diagonal elements are non-zero, it is called identity matrix and denoted by I.
Equal Matrices: Two matrices are said to be equal if they are of the same order and if their corresponding elements are equal to the square matrix A = [aij]n × n is an identity matrix if
aij = 1 if i = j
aij = 0 if i ≠ j
We denote the identity matrix of order n by In. When the order is clear from the context, we simply write it as I.
9) Upper Triangular Matrix
A square matrix in which all the elements below the diagonal are zero is known as the upper triangular matrix.
10) Lower Triangular Matrix
A square matrix in which all the elements above the diagonal are zero is known as the upper triangular matrix.