Operations on Matrices

Matrix is an essential concept of mathematics that involves several algebraic operations to carry out complex and simple mathematics problems and other physical applications in a simple manner. The different functions in the matrix concept include basic algebraic operations like adding matrices, subtraction, and multiplication of a matrix by a scalar. These matrix operations help us simplify complex problems and algorithms of mathematics and other physical applications. This article will provide you with a comprehensive piece of knowledge on several matrix operations like multiplication of a matrix by scalar and the properties of these operations in a detailed manner. 

Different types of Matrices 

Now you have understood and become familiar with the concept of matrices in mathematics. Let’s move on to understand the different types of matrices in mathematics. In general, there are six types of matrices, which are categorised by the values of the elements. Below mentioned are six different types of matrices in mathematics:

1. Square Matrix 

2. Diagonal Matrix 

3. Triangular Matrix

4. Identity Matrix

5. Symmetric Matrix

6. Orthogonal Matrix 

Different operations on Matrices 

We have covered the basic concept of the matrix. You have become quite familiar with the different types of Matrices in mathematics. Let’s move on to the other operations you can perform on any matrix and utilise it to solve various problems like linear algebra, quadratic equations, etc.

There are four basic operations in Matrices based on their nature. They are named as:

• Addition of Matrices

• Subtraction of Matrix 

• Multiplication of Matrix 

• Scalar Multiplication of Matrix 

Before moving on to the detailed information on each of these operations, you must note that they should have an exact order to apply multiplication operations on any matrices simple words, the number of rows of the second matrix should be equal to the number of columns of the first matrix to apply the operation of multiplication on any two matrices. 

Addition of Matrices

If there are two matrices A and B of same order 3×3, then the sum of both the matrices A and B will equal to the sum of corresponding elements of both the matrices. In simple words if the elements of matrix A are denoted as a11, a12, a13, a21, a22, a23, a31, a32, a33 and the elements of matrix B are denoted as b11, b12, b13, b21, b22, b23, b31, b32, b33. The sum of the matrices A and B will be a matrix C of the same order and elements of the C matrix will be c11, c12, c13, c21, c22, c23, c31, c32, c33. The values of these elements will be calculated as given below

c11 = a11+b11

c12 = a12+b12

c13 = a13+b13

c21 = a21+b21

c22 = a22+b22

c23 = a23+b23

c31 = a31+b31

c32 = a32+b32

c33 = a33+b33

Subtraction of Matrices 

If there are two matrices, A and B, of the same order 3×3, the difference of both the matrices A and B will equal the corresponding elements of both the matrices.  In simple words if the elements of matrix A are denoted as a11, a12, a13, a21, a22, a23, a31, a32, a33 and the elements of matrix B are denoted as b11, b12, b13, b21, b22, b23, b31, b32, b33. The difference of the matrices A and B will be a matrix C of the same order and elements of the C matrix will be c11, c12, c13, c21, c22, c23, c31, c32, c33. The values of these elements will be calculated as given below

c11 = a11-b11

c12 = a12-b12

c13 = a13-b13

c21 = a21-b21

c22 = a22-b22

c23 = a23-b23

c31 = a31-b31

c32 = a32-b32

c33 = a33-b33

Multiplication of Matrix 

Suppose there are two matrices, C and D, then carry out the multiplication operation. In that case, the number of columns of C should be equal to the number of rows of D.

Now, if C is defined as [cij]m×n and D is defined as [dij]n×k, then the product of matrices C and D will be equal to P= CD= [Pij]m×k

Scalar Multiplication of Matrix 

As the name suggests, the scalar multiplication of matrices refers to the expansion of any scalar quantity ‘K’ with all the matrix elements. In simple words, if there is a matrix X which is defined as [xij]m×n, and it is multiplied by a scalar K, then the resultant matrix will be KXm×n= Xm×n K = [Kxi×j].

Conclusion 

Matrix is fun to do part of mathematics that can be defined as combining elements in a rectangular array. These elements undergo several operations to solve several complex problems more straightforwardly. There are four basic operations: addition, subtraction, multiplication, and scalar multiplication. This article comprehensively discusses all these operations in detail to fully understand different functions possible on matrices.