Hey there, in this post, we’ll discuss and know about the most interesting and well known topic- ‘Matrices’along with its operations & types. Matrices have wide scope in mathematics as well as in physics. Let us know about Introduction to matrices and their operations…
Introduction to matrices
Definition of Matrices:
A rectangular arrangement of ‘mn’ numbers in ‘m’ rows and ‘n’ columns, each row contains ‘n’ numbers so as to form ‘n’ columns. Whole system is enclosed between rectangular brackets is called a matrix of order m×n.
The numbers which form a matrix are known as ‘elements’ of that matrix. A matrix is denoted by capital letters such as matrix A,B,C,… etc.and its elements are denoted by corresponding small letters with two suffixes. An element at position ‘i’ th row and ‘j’th column, is called ‘ij’ element of the corresponding matrix. it is denoted by ‘aij‘thus, for example if ‘a’ is an element at the position of third row and second column then it can be denoted as ‘a32‘.The first suffix denotes row and the second suffix denotes its corresponding column respectively.
a11 = 5 a12 = 3 a13 = 2 a14 = -5 a21 = 3 a22 = 0 a23 = -1 a24 = 10 a31 = -3 a32 = 9 a33 = 15 a34 = 20
Order of Matrices:
Order of matrices is the notation to understand the dimension of the matrix. It is given by the product of row and columns.
Consider for example, in above matrix, the order of matrix is 3 × 4 i.e. there are three rows and four columns in above matrix.
Let us now discuss the types of the matrices and their corresponding properties…
Types of matrices:
Definition: A square matrix is a matrix in which the no. of rows is same as the number of columns. In other words, the square matrix can be defined as the matrix of order ‘n×n’. (say,’3×3′ or ‘1×1’ etc.)
This is a 4×4 matrix.
Definition: A matrix which is not square is a rectangular matrix. A matrix of order ‘n×m’ where m ≠ n
consider a matrix in the definition of matrix, this is also a rectangular matrix.
Definition: A matrix having only one row is a row matrix.
For example: matrix below is a row matrix of the order ‘1×3’
Definition: A matrix having only one column is a column matrix.
For example: matrix below is a row matrix of the order ‘3×1’
Zero Matrix / Null Matrix:
Definition: A matrix each of whose elements are zeros, is called a zero matrix or null matrix.
A zero matrix of order ‘m×n’ is denoted by Omn
For example: matrix below is a zero matrix of the order ‘4×1’
Definition: Let A be any square matrix of order n×n, then all its ‘aij‘ elements are ‘diagonal elements’,where i=j.
For example: Consider the matrix below, here the diagonal elements are,3, 5, 1.
A square matrix in which all non diagonal elements are zero is called a diagonal matrix.
For example: Consider the matrix below, is a Diagonal Matrix.
A diagonal matrix in which all diagonal elements are same, is a scalar matrix.
Unitary Matrix is also a Scalar Matrix.
Unitary Matrix or Identity Matrix:
A Scalar matrix in which all diagonal elements are equal to one (1), is an identity matrix.
A unitary matrix of order n is denoted by ‘I’ or’In‘
For example: Consider the matrix below,
Upper Triangular Matrix:
A Square matrix is called an Upper triangular matrix if all the elements below the diagonal are zero.
For example: Consider the matrix below, it is an upper triangular matrix of order ‘3×3’
Lower Triangular Matrix:
A Square matrix is called an Lower triangular matrix if all the elements above the diagonal are zero.
For example: Consider the matrix below, it is an lower triangular matrix of order
Here, we take a short break in types of matrices,first we see how to find inverse of the matrices, how to solve determinants, because the further types are depend upon the inverse of the matrices.
Evaluation of the Determinants of the Matrices:
Determinant plays an important role in finding the inverse of the Matrix and also it confirms the singularity of matrices. Let us now see how to find determinant of given matrix:
Determinants are denoted by |A| or det. A. To find determinant there are simple rules:
For ‘2×2’ Matrix,
Determinant above matrix= a1 b2 – a2 b1
For ‘3×3’ matrix:
Determinant above matrix= a1 [b2 c3 – b3 c2] – b1[ a2c3 – a3 c2] + c1[b3 a2 – b2 a3]
- If the determinant of the matrix is zero, then it is said to be singular matrix or in other words, it is said to be non-invertible i.e. the inverse of the matrix does not exist.
- If the determinant of the matrix is non-zero, then it is said to be non-singular matrix or in other words, it is said to be invertible i.e. the inverse of the matrix exist.
- i.e. Singularity & invertiblity of matrices are opposite properties.
Now, we learn about the properties and the operations on the matrices:
Equality of Matrices:
Two matrices A & B are said to be equal if A&B are of the same order, and each element of matrix A is same to the corresponding element of the B matrix, then we can write A = B.
Operations on Matrices:
Transpose of the Matrix:
A matrix A’ is obtained by interchanging rows and columns of the matrix A is called the transpose of the matrix A. It is denoted by A’.
- If the order of the matrix A is ‘m×n’ then, the order of the transposed matrix A’ is ‘n×m’.
- Also, the aijth element of matrix A is equal to the ajith element of matrix A’.
- The transpose of the transposed matrix is the original matrix itself. i.e. (A’)’=A .
Multiplication of a matrix by a scalar:
In the study of matrices,the numbers called scalars. the multiplication operations of matrices is the multiplication of each element of the matrix by a given scalar. Suppose the given matrix is multiplicated by a scalar 3, then it gives a new matrix with different elements.
If the given matrix is multiplicated by (-1) then it would be the negative of the given matrix.
Addition of the Matrices:
Let A & B be any two matrices of the same order. The sum of A & B is denoted by A+B.It is defined to be the matrix obtained by adding A&B.
- One thing must be remembered that, for the addition of two matrices they must be of same order.
- Matrix addition is commutative, i. e. if A & B are two matrices of same order
- If A,B & C are three matrices of same order, then (A+B)+C=A+(B+C).
We consider an example to illustrate the matrix addition:
Difference of two Matrices:
Let A & B be two Matrices of same order. Then their difference operations is denoted by A-B. It is defined to be the matrix A+(-B).
We consider an example to illustrate the difference:
We’ll discuss in next post…