Question

Types of Matrices

Answer 1

Here, we shall discuss different types of matrices.

1. Rectangular Matrix : A matrix in which number of rows are not equal to number of columns, is called rectangular matrix.

There are two types of rectangular matrix :

(i) Horizontal Matrix : A matrix in which number of columns are greater than number of rows is called horizontal matrix.

Here, number of columns = 3 and number of rows = 2

(ii) Vertical Matrix : A matrix in which number of rows are greater than number of columns, is called vertical matrix.

Here, number of rows = 4 and number of columns = 3

2. Square Matrix : A matrix in which the number of rows are equal to the number of columns, is said to be square matrix. Thus, m x n matrix is said to be a square matrix where m = n and is known as a square matrix of order V.

3. Diagonal Matrix : A, square matrix A = [a_ij]_{m × n} is called a diagonal matrix, if all the elements excepts those in the leading diagonal, are zero, i.e., a_ij = 0 for all i ≠ j. A diagonal matrix of order n × n having d₁, d₂, ................ d_n as diagonal element is denoted by

diag [d₁, d₂, ...................... d_n]

4. Row Matrix: A matrix having only one row is called a row matrix.

Example: A = \([-\frac{1}{2}\sqrt5\ 2\ 3]_{1 \times 4}\) is a row matrix of order 1 × 4.

5. Column Matrix: A matrix having only one column is called column matrix.

6. Scalar Matrix: A diagonal matrix is said to be a scalar matrix if its diagonal elements are equal, that is, a square matrix A = [a_ij]_{m × n} is called a scalar matrix if

• a_ij = 0 for all i ≠ j and
• a_ij = c for all i = j, where c ≠ 0

are scalar matrices of order 1,2 and 3 respectively.

7. Identity or Unit Matrix : A square matrix in which elements in the diagonal are all 1 and rest are all zero is called an identity matrix. In other words, a square matrix A = [a_ij] is called an identity or unit matrix if

• a_ij = 0 for all i ≠ j and
• a_ij = 1 for all i = j

The identity matrix of order rt is denoted by I_n.

are identity matrices of order 1, 2, 3 respectively.

8. Null Matrix or Zero Matrix: A matrix is said to be null matrix or zero matrix if all its elements are zero.

9. Triangular Matrix

(i) Upper Triangular Matrix: A square matrix A = [a_ij] is called an upper triangular matrix if = 0 for all i > j matrix.

(ii) Lower Triangular Matrix : A square matrix A = [a_ij] is called a lower triangular matrix if a_ij = 0 for all i < j.
Thus, in a lower triangular matrix all elements above the main diagonal are zero.

A triangular matrix A = [ai_ij]_{n × n} is called a strictly triangular if a_ij = 0 for all i = 1, 2......... n

10. Sub-matrix: If some row and columns are deleted from a matrix, then remaining matrix is called sub-matrix of given matrix.

11. Comparable Matrices : Two matrices A and B are said to be comparable if they are of same order.

Here A and B are comparable matrices, since their order is 2 × 3 (same).