Types of Matrices
In this section, we shall define certain types of matrices.
A matrix is said to be a row matrix if it has only one row and any number of columns.
A row matrix is also called as a row vector.
For example, A = (8 9 4 3), B = (-√3/2 1 √3) are row matrices of order 1×4 and 1×3 respectively.
In general A = (a11 a12 a13 … a1n) is a row matrix of order 1×n.
A matrix is said to be a column matrix if it has only one column and any number of rows. It is also called as a column vector.
For example, are column matrices of order 3 ×1, 2 ×1 and 4 ×1 respectively.
In general, is a column matrix of order m × 1.
A matrix in which the number of rows is equal to the number of columns is called a square matrix. Thus a matrix A = (aij )m ×n will be a square matrix if m = n
For example, are square matrices.
In general, are square matrices of orders 2 × 2 and 3 × 3 respectively.
A = (aij )m ×m is a square matrix of order m.
Definition : In a square matrix, the elements of the form a11, a22, a33, . . . (i.e) aii are called leading diagonal elements. For example in the matrix , 1 and 5 are leading diagonal elements.
A square matrix, all of whose elements, except those in the leading diagonal are zero is called a diagonal matrix.
(ie) A square matrix A = (aij ) is said to be diagonal matrix if aij = 0 for i ≠ j . Note that some elements of the leading diagonal may be zero but not all.
For example, are diagonal matrices.
A diagonal matrix in which all the leading diagonal elements are equal is called a scalar matrix.
In general, A = (aij )m ×m is said to be a scalar matrix if
A square matrix in which elements in the leading diagonal are all “1” and rest are all zero is called an identity matrix (or) unit matrix.
Thus, the square matrix A = (aij) is an identity matrix if aij =
A unit matrix of order n is written as In.
are identity matrices of order 2 and 3 respectively.
A matrix is said to be a zero matrix or null matrix if all its elements are zero.
For example, are all zero matrices of order 1×1, 2 × 2 and 3 × 3 but of different orders. We denote zero matrix of order n ×n by On.
is a zero matrix of the order 2×3.
The matrix which is obtained by interchanging the elements in rows and columns of the given matrix A is called transpose of A and is denoted by AT (read as A transpose). For example,
If order of A is m ×n then order of AT is n ×m .
We note that (AT )T = A.
A square matrix in which all the entries above the leading diagonal are zero is called a lower triangular matrix.
If all the entries below the leading diagonal are zero, then it is called an upper triangular matrix.
A square matrix A = (aij )n ×n is called upper triangular matrix if aij = 0 for = 0 for i > j and is called lower triangular matrix if aij = 0 , i < j .
For example, A = is an upper triangular matrix and B = is a lower triangular matrix.
Two matrices A and B are said to be equal if and only if they have the same order and each element of matrix A is equal to the corresponding element of matrix B. That is, a ij = bij for all i, j.
For example, if
then we note that A and B have same order and aij = bij for every i, j. Hence A and B are equal matrices.
1. The number of column(s) in a column matrix are _______.
2. The number of row(s) in a row matrix are _______.
3. The non-diagonal elements in any unit matrix are______.
4. Does there exist a square matrix with 32 elements?
The negative of a matrix Am ×n denoted by −Am ×n is the matrix formed by replacing each element in the matrix Am ×n with its additive inverse.
Additive inverse of an element k is -k . That is, every element of –A is the negative of the corresponding element of A.
For example, if
Consider the following information regarding the number of men and women workers in three factories I, II and III.
Represent the above information in the form of a matrix. What does the entry in the second row and first column represent?
The information is represented in the form of a 3 ×2 matrix as follows
The entry in the second row and first column represent that there are 47 men workers in factory II
If a matrix has 16 elements, what are the possible orders it can have?
We know that a matrix of order m ×n , has mn elements. Thus to find all possible orders of a matrix with 16 elements, we will find all ordered pairs of natural numbers whose product is 16.
Such ordered pairs are (1,16), (16,1), (4,4), (8,2), (2,8)
Hence possible orders are 1 × 16, 16 ×1 , 4 ×4 , 2 ×8 , 8 ×2
Construct a 3 ×3 matrix whose elements are a ij = i2 j 2
The general 3×3 matrix is given by
Find the value of a, b, c, d from the equation
The given matrices are equal. Thus all corresponding elements are equal.
a − b = 1…(1)
2a + c = 5…(2)
2a − b = 0…(3)
3c + d = 2…(4)
2a − b = 0
2a = b …(5)
Put 2a = b in equation (1), a − 2a = 1 gives a = −1
Put a = −1 in equation (5), 2(− 1) = b gives b = −2
Put a = −1 in equation (2), 2(− 1) +c = 5 gives c = 7
Put c = 7 in equation (4), 3(7) + d = 2 gives d = −19
Therefore, a = −1, b = −2, c = 7, d = −19