Matrix: Definition & Types of Matrix

Welcome to you guys. Today we are going to learn a very interesting topic, Matrix.

So, what are you waiting for? Just put everything aside and concentrate on this page.

let’s start-

1. What is a Matrix?

⇒ A Matrix is nothing but an arrangement of some numbers into rows and numbers. We present the matrix in the sign of [ ] or ( ). For example,

So, as you can see here, Matrix looks like this. You can also use (). Now let’s know about the history of the matrix.

2. History:-

The term matrix was introduced in the 19th century by an English mathematician James Sylvester and his friend Arthur Cayley developed the algebraic aspect of the matrix. Cayley first applied them to the study of the system of linear equations. Cayley recognized that certain sets of matrices follow many of the laws of arithmetic, like- associative, distributive, etc, and also deny some other laws like- commutative law. But the important thing is that was not the matrix but the determinant which was first recognized for solving the problems of linear equations.

Order of a Matrix:

The order of a matrix shows us how many rows and columns are there in the given matrix. We generally denoted it by simply m⨯n. And the sign ⨯ is called ‘by’. Here m denotes the number of columns and n denotes the number of rows.

Now let’s see some

Types of Matrixes:

1.Rectangular Matrix:

If the numbers of rows(m) are not equal to the numbers of columns(n), then that type of matrix is called the Rectangular matrix.

2. Square Matrix:

A square matrix is a matrix in which the number of rows and columns are the same. We usually denote these matrixes by

(A_ij)_{n x n}
Just like the example.

Now the square matrix is of two types. They are-

Singular Matrix:

A Matrix is said to be singular if the determinant of a square matrix becomes zero i.e. |A_ij|=0

Non-singular Matrix:

Clearly, if the determinant of a square matrix is a nonzero term then the matrix will be nonsingular, i.e. |A_ij>|≠0

3.Row column matrix:

Row matrix:

The matrix which contains only rows, not the column is said
Row matrix. i.e. in the given example n=0.

Column Matrix: The matrix which contains only columns, not the row is said Column Matrix.

4. Null Matrix:

5.Diagonal Matrix:

e.g.=

6. Scalar Matrix:

7. Identity Matrix:

8.Upper triangular and lower triangular Matrix:

First, If in a square matrix, the elements along the starting diagonal are zero, then this type of matrix is called Triangular Matrix.

In which square matrices, the elements along the starting diagonal are not zero, but if the elements below the triangle that divides by the diagonal the square matrix into two triangular parts are zero, those matrices are called Upper triangular Matrix.

But if the elements on the upper triangle are zero, then it should be the Lower triangular Matrix.

9. Transpose of a Matrix:

For example-

So, from this image, we can see that where A is a 2⨯3 matrix, A’ is a 3⨯2 matrix as usual.

10. Symmetric & Skew-symmetric Matrix:

If the transpose of a  square matrix is the same matrix(with the same sign), i.e A’=A, then this type of matrix is called Symmetric Matrix.

And if the transpose matrix is the same but the sign is negative, i.e A’=-A, then it is called Skew-symmetric Matrix.

Now, let’s talk about other properties of Matrix. Like-

1. Multiplication of a matrix:

First of all, there are two types of multiplication of Matrix.

1) Scalar Multiplication of a matrix:

From the heading, hope you can clearly understand that if there is a matrix A and a scalar quantity n, then their scalar multiplication will be like- 

 and denoted by nA.(I hope the way I give the example is not making any problem for you to understand.)

2) Multiplication of two Matrices:

First, 

Let, A and B be two matrices of order 2⨯2 ( I’m considering 2⨯2 because it will help you to understand the process more smoothly). 

Now, if we want to multiply them then this will seems like this-

Hope you understand this way.

As you can see here, the process is the same. But, look carefully, the order of the resultant matrix is changed.

So, as you can see that this time the order is 2⨯2 of the resultant matrix.

Now let’s jump up to the next property.

Addition and Subtraction of Matrices:

Rules:

Addition and Subtraction of two or more matrices are only possible if and only if they are of the same order. For example,

Suppose we are adding two matrices A and B. Now this means that they both are of the order m⨯n. And m,n is the same for both matrices.

So, in this example, you can see the result. And this will happen in every case. 

Now here comes some characteristics of this property.

If A, B, C are three matrices of the same order then,

1. A+B=B+A, which means it satisfies “Commutative Law”.

2. (A+B)+C=A+(B+C), i.e it satisfies “Distribution Law”.

3. k(A+B)=kA+kB, where k is an arbitrary constant.

4.A+0=0+A=A, where 0 is actually the Zero-Matrix of the same order as A.

5. If, A+B=B+C then, A=C

Zero-Matrix:

This is a kind of matrix where each and every element are zero.

Null Matrix:

A matrix is called a null matrix if there is no element in the matrix.

Ok, understood?

Great. 

Now let’s jump up to the next property, which is-

We read before that we denote the transpose matrix of A as A^T or A’. So, here is the next property-

Orthogonal Matrix:

So, if AA’=A’A=I(Identity matrix), then A is called an Orthogonal Matrix.

Now we talk later about orthonormal vectors and all. Because before entering that we have to know some more things.

So, for now, let’s talk about the characteristics of the Transpose matrix.

If A’ and B’ are the transpose of the matrices A and B then,

1.(A’)’=A

2.(A+B)’=A’+B’

3. (A-B)’=A’-B’

4. (AB)’=B’A’

Great, now see what’s next-

Sum of Symmetric and skew-symmetric:

We can represent a square matrix A as the sum of the symmetric and the skew-symmetric matrix like-

A=(A+A’)/2 +(A-A’)/2, where (A+A’)/2 is the symmetric matrix and (A-A’)/2 is the skew-symmetric matrix.

Now, that’s enough for this.

We will jump up to some more things about Matrix in the next part.

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Operations:

1. Row Operations:

I hope you all know these. Despite that, let’s take a look at this.

Let A be a matrix of order m⨯n.

Now, in a row operation, we can do these three operations.

1. We can interchange the rows.

2. We can multiply any row of A with a nonzero constant.

3. We can add a row to another which is multiplied by a number or can subtract them.

For example-

2. Column Operation:

Similarly,

 Let A be a matrix of order m⨯n.

1. We can interchange the columns.

2. We can multiply any column of A with any nonzero constant.

3. We can add a column to another which is multiplied by a number or can subtract them.

Similarly,

Equivalent Matrix:

If we get a matrix by row and column operation on another matrix, then the former matrix is called the Equivalent Matrix of the latter one.

Now, there are two types of equivalent matrix.

1.Row Equivalent Matrix:

If a matrix is coming from another matrix by only row operation, then the matrix is called the Row Equivalent matrix.

For example, let a 3⨯3 matrix is A and another 3⨯3 matrix is B.

2. Column Equivalent Matrix:

Similarly, if a matrix is coming from another matrix by only column operation, then the matrix is called Column Equivalent Matrix.