Let us first try to understand what a matrix is and why we use it? 

All the elements are represented in a rectangular or square form in the matrix. Matrices help perform operations like addition, subtraction, multiplication, division, transpose, and the inverse. It is a representation of numbers arranged in rows and columns. The horizontal lines of matrices are called columns, and the vertical lines are called rows. The size of a matrix is calculated by the number of rows and columns it has. A matrix with m rows and n columns is called an m × n matrix, while m and n are called its dimensions. There are six different types of matrices in mathematics. 

Square Matrices: A matrix containing the same number of columns and rows is called a square matrix. Example – 2×2 matrix, 3×3 matrix, 4×4 matrix, etc. These matrices look like a perfect square having equal sides. Hence, they are called a square matrix.

Rectangle Matrices: A matrix containing an unequal number of rows and columns is called a rectangular matrix. For example – 2×3,4×7, 2×1, etc.

History of Matrix

In 1850, James Joseph Sylvester coined the word matrix. He comprehended matrix as an item giving several determinants, also called minor. The word matrix was derived from the Latin word “womb.” Matrices are used in the key generation for encryption and decryption. They are also used for object altering in 3D games.

Different Types of Matrices

The six types of matrices are given below:

1. Diagonal Matrix 
2. Square Matrix
3. Triangular Matrix
4. Symmetric Matrix 
5. Identity Matrix 
6. Orthogonal Matrix 

Multiplying matrices of 2×2

We store elements in matrices to perform operations like addition, subtraction, multiplication, division, etc. In this section, we will learn the multiplying matrices of 2×2. One should know the concept of multiplying the matrices of 2×2 as it is a very important topic in matrices.

There are two types of multiplying matrices of 2×2. The first one is scalar multiplication, and another one is matrix multiplication.

1. In scalar multiplication, each element in the matrix is multiplied by a single number. Each element in the matrix is multiplied by that number, and the result is also scalar.

1. Another one is Matrix multiplication. It is also known as the dot product. In dot product, two matrices are multiplied to obtain a result which is also a matrix. For example, if we multiply matrix X and matrix Y of 2×2 order, the result will be Matrix Z of 2×2 order only. 

Example for multiplying matrices of 2×2: A.B= C

Note: 

1. Do not use multiplication sign (x) in matrix multiplication representation. Remember it as this silly mistake in deducting the marks. Use dot sign (.) only as shown in the above example.
2. Matrix multiplication is different from ordinary multiplication methods and scalar methods. Hence, do not try to simply multiply the corresponding elements as usually done by addition and subtraction of matrices.
3. The resulting matrix will be of 2×2 dot product, if and only if two matrices are of order 2×2. Hence, if you found any other order during multiplying matrices of 2×2, then rectify the mistake immediately.

Now let us see the steps to solve the dot product of the matrix of 2×2 or how to multiply matrices 2×2.

Steps:

1. Ensure that the number of elements in the first column of matrix A and the number in the first row of matrix B is equal. If the number of elements in the first column of matrix A and the first row in matrix B is not equal, the dot product is impossible. But in matrices of order 2×2, the above condition is automatically satisfied.
2. Then start multiplying the elements of each row of matrix A by the elements of each column of matrix B. After multiplying all the elements, add them for a whole number.

Multiplying matrices of 3×3

Now let us see how to multiply matrices of 3×3.

Steps: 

1. Ensure that the number of elements in the first column of matrix A and the number in the first row of matrix B is equal if the number of elements in the first column of matrix A and the first row in matrix B is not equal.
2. Then start multiplying the elements of each row of matrix A by the elements of each column of matrix B. After multiplying all the elements, add them for a whole number.

Determinants of Matrices

Determinant of Matrix is an operation on the elements of a matrix to obtain a single number by the sum of the products of the elements present in the matrix. It has extensive use in the area of measurement. While solving linear equations will tend to use determinant methods. The determinant method is easy and hassle-free for solving linear equations. Also, while solving integrals, the determinant of the matrix is used. Also, determinants can be used to find out the area and volumes of the triangle, parallelograms and many other shapes. The determinant of the matrix is very useful in mensuration.

Conclusion

In this article, we learned some topics related to matrices and determinants. We looked after the history and use of matrices. We studied both methods of multiplying 2×2 matrices in steps. Also, we learnt about the types of matrices and solution of matrices. It was all about matrices, determinants and matrix multiplication of 2×2 and 3×3 matrices. I hope you will understand all these topics after reading this article.