The different types of matrices are:
Row Matrix
A matrix having only one row is called a row matrix. Thus A = [a_{ij}]_{mxn} is a row matrix if m = 1. So, a row matrix can be represented as A = [a_{ij}]_{1×n}. It is called so because it has only one row and the order of a row matrix will hence be 1 × n. For example, A = [1 2 4 5] is row matrix of order 1 x 4. Another example of the row matrix is P = [ 4 21 17 ] which is of the order 1 × 3.
Column Matrix
A matrix having only one column is called a column matrix. Thus, A = [a_{ij}]_{mxn} is a column matrix if n = 1. Thus, the value of for a column matrix will be 1. Hence, the order is m × 1.
An example of a column matrix is:
is column matrix of order 4 x 1.
Just like the row matrices had only one row, column matrices have only one column. Thus, the value of for a column matrix will be 1. Hence, the order is m × 1. The general form of a column matrix is given by A = [a_{ij}]_{m×1}. Other examples of a column matrix include:
In the above example , P and Q are 3 ×1 and 5 × 1 order matrices respectively.
Zero or Null Matrix
If in a matrix all the elements are zero then it is called a zero matrix and it is generally denoted by 0. Thus, A =
[a_{ij}]_{mxn} is a zeromatrix if a_{ij} = 0 for all i and j; E.g. is a zero matrix of order 2 x 3.
is a 3 x 2 null matrix & B = is 3 x 3 null matrix.
Singleton Matrix
If in a matrix there is only element then it is called singleton matrix. Thus, A = [a_{ij}]_{mxn} is a singleton matrix if m = n = 1. E.g. [2], [3], [a], [] are singleton matrices.
Horizontal Matrix
A matrix of order m x n is a horizontal matrix if n > m; E.g.
Vertical Matrix
A matrix of order m x n is a vertical matrix if m > n; E.g.
Square Matrix
If the number of rows and the number of columns in a matrix are equal, then it is called a square matrix.
Thus, A = [a_{ij}]_{mxn} is a square matrix if m = n; E.g.
is a square matrix of order 3 × 3.
The sum of the diagonal elements in a square matrix A is called the trace of matrix A, and which is denoted by
Another example of a square matrix is:
The order of P and Q is 2 × 2 and 3 × 3 respectively.
Diagonal Matrix
If all the elements, except the principal diagonal, in a square matrix, are zero, it is called a diagonal matrix. Thus, a square matrix A = [a_{ij}] is a diagonal matrix if a_{ij} = 0,when i ≠ j; E.g.
is a diagonal matrix of order 3 x 3, which can also be denoted by diagonal [2 3 4]. The special thing is, all the nondiagonal elements of this matrix are zero. That means only the diagonal has nonzero elements. There are two important things to note here which are
(i) A diagonal matrix is always a square matrix
(ii) The diagonal elements are characterized by this general form: a_{ij} where i = j. This means that a matrix can have only one diagonal.
Few more example of diagonal matrix are:
P = [9]
In the above examples, P, Q, and R are diagonal matrices with order 1 × 1, 2 × 2 and 3 × 3 respectively. When all the diagonal elements of a diagonal matrix are the same, it goes by a different name scalar matrix which is discussed below.
Scalar Matrix
If all the elements in the diagonal of a diagonal matrix are equal, it is called a scalar matrix. Thus, a square matrix A = [a_{ij}]_{mx }is a scalar matrix if a_{ij} = where k is a constant.
is a scalar Matrix.
More examples of scalar matrix are:
Now, what if all the diagonal elements are equal to 1? That will still be a scalar matrix and obviously a diagonal matrix. It has got a special name which is known as the identity matrix.
Unit Matrix or Identity Matrix
If all the elements of a principal diagonal in a diagonal matrix are 1, then it is called a unit matrix. A unit matrix of order n is denoted by I_{n}. Thus, a square matrix A = [a_{ij}]_{m×n} is an identity matrix if
Conclusions:
It should be noted that the converse of the above statements is not true for any of the cases.
Equal matrices are those matrices which are equal in terms of their elements. The conditions for matrix equality are discussed below.
Equality of Matrices Conditions
Two matrices A and B are said to be equal if they are of the same order and their corresponding elements are equal, i.e. Two matrices A = [a_{ij}]_{m×n} and B = [b_{ij}]_{r×s} are equal if:
(a) m = r i.e. the number of rows in A = the number of rows in B.
(b) n = s, i.e. the number of columns in A = the number of columns in B
(c) a_{ij} = b_{ij}, for i = 1, 2, ….., m and j = 1, 2, ….., n, i.e. the corresponding elements are equal;
For example, Matrices are not equal because their orders are not the same.
But, If
are equal matrices then,
a_{1} = 1, a_{2} = 6, a_{3} = 3, b_{1} = 5, b_{2} = 2, b_{3} = 1.
A square matrix is said to be a triangular matrix if the elements above or below the principal diagonal are zero. There are two types:
Upper Triangular Matrix
A square matrix [a_{ij}] is called an upper triangular matrix, if a_{ij} = 0, when i > j.
is an upper uriangular matrix of order 3 x 3.
Lower Triangular Matrix
A square matrix is called a lower triangular matrix, if a_{ij} = 0 when i < j.
is a lower triangular matrix of order 3 x 3.
Matrix A is said to be a singular matrix if its determinant A = 0, otherwise a nonsingular matrix, i.e. If for det A = 0, it is singular matrix and for det A ≠ 0, it is nonsingular.
Symmetric and Skew Symmetric Matrices
Note: A is symmetric if A’ = A (where ‘A’ is the transpose of matrix)
Thus, in a skewsymmetric matrix all diagonal elements are zero; E.g.
are skewsymmetric matrices.
Note: A square matrix A is a skewsymmetric matrix A’ = A.
Some important Conclusions on Symmetric and SkewSymmetric Matrices:
A square matrix A = [a_{ij}] is said to be a Hermitian matrix if a_{ij}
are Hermitian matrices
Important Notes:
A = [a_{ij}] is said to be a skewHermitian if i.e. A^{θ} = – A;
are skewHermitin matrices.
i.e. a_{ii} must be purely imaginary or zero.
Special Matrices
(a) Idempotent Matrix: A square matrix is idempotent, provided A^{2} = A. For an idempotent matrix
For an idempotent matrix A, det A = 0 or x.
(b) Nilpotent Matrix: A nilpotent matrix is said to be nilpotent of index p, (p ∈ N), if A^{p} = O, A^{p  1} ≠ O, i.e. if p is the least positive integer for which A^{p} = O, then A is said to be nilpotent of index p.
(c) Periodic Matrix:
A square matrix which satisfies the relation A^{k + 1} = A, for some positive integer K, then A is periodic with period K, i.e. if K is the least positive integer for which A^{k + 1} = A, nd A is said to be periodic with period K. If K =1 then A is called idempotent.
has the period 1.
Notes:
(i) Period of a square null matrix is not defined.
(ii) Period of an idempotent matrix is 1.
(d) Involutary Matrix:
If A^{2} = I, the matrix is said to be an involutary matrix. An involutary matrix its own inverse
184 videos552 docs187 tests

1. What is a matrix? 
2. What are the different types of matrices? 
3. What is the significance of a square matrix? 
4. How is an identity matrix useful in matrix operations? 
5. How can symmetric matrices be utilized in realworld applications? 
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