Equality of matrices mathematics

What is equality of matrices?

A rectangular array of numbers or expressions arranged in rows and columns is called a matrix. When two or more matrices are equal with the same number of rows, columns and other elements, it is defined as equality of matrices.

Equality of matrices is a concept found in both rectangular and square matrices. The matrices are not equal if they do not hold the mentioned properties. In other words, if the order of a pair of elements is not equal to the elements corresponding to them, then the matrices will not be equal. In this equality of matrices study material, we will solve some examples to gain a better understanding of the concept.

What are the conditions for matrix equality?

There are three conditions required for the equality of matrices. Let us take an example, 

A = [aij]m×n and B = [bij]p×q :

• Matrices A and B should have the same number of rows, that is, m =p 
• Matrices A and B should have the same number of columns. that is m=q 
• The corresponding elements of A and B should be equal. that is aij=bij 

To know more about matrices, continue reading this study material notes on the equality of matrices.

 Let us look at an example of two matrices. Consider row matrices A = [1 2 x]1×3 and B = [y 2 7 ]1×3. Both matrices A and B have the same number of rows and the same number of columns. So they have the same dimensions. If these matrices are equal, their corresponding elements will also be equal. So, 1=y, 2=2, and x=7.

How to solve a matrix?

Here is an example:

[3x+4y  x−2y   6] = [ 2 4 6]

  [a+b  −3  2a−b]=[ 5 −3 −5]

The order of the matrices is equal, thus the corresponding element will also be equal. therefore, we have:

⇒ 3x + 4y = 2 ……(1)

x – 2y = 4 ……(2)

a + b = 5……(3)

2a – b = -5 ……(4)

Solving equations (1) and (2), we have

x = 2y + 4 

Substituting the above equation in (1),

3(2y + 4) + 4y = 2

6y + 12 + 4y = 2

10y = 2 – 12

10y = -10

y = -1

So, x = 2(-1) + 4

= -2 + 4

= 2

Similarly, we solve equations (3) and (4), we have a = 0 and b = 5.

Types of matrices

There are six types of matrices;

Square matrix 

It is a matrix where several rows equal several columns. Since the rows and columns match, the dimensions denote N. Order is the size of the matrix, so order 4 square matrix is 4 x 4.

Symmetric matrix

It is a square type matrix where the top-right triangle is the same as the bottom-left triangle. For the matrix to be symmetrical, the axis of symmetry is the main diagonal, from top left to bottom right. 

Triangular matrix 

It is a type of square matrix, with all values in the upper right or lower left of the matrix while other elements are zero. An upper triangular matrix is a triangular matrix with values above the main diagonal. A lower triangular matrix is a triangular matrix with values below the main diagonal. 

Diagonal matrix 

It is a matrix where values outside the main diagonal have zero value. The main diagonal is from the top left to the bottom right of the matrix. The diagonal matrix can be both a square as well as a rectangular matrix. 

Identity matrix 

It is a square matrix whose value does not change when a vector multiplies. The main diagonal is from top left to bottom right, having the value one, while all other values are zero. 

Orthogonal matrix

It is where two vectors are orthogonal when their dot product equals zero. It is a type of square matrix where the rows and columns are vectors.

Properties of matrices (addition) 

A matrix can be added with another if the order of matrices is the same. The addition takes place between the elements. The resultant matrix will also have the same order. For example, 

[ 1 4 ]+ [4 6] = [ 5 10]    

  [5 3 ]+[ 7 2]=[ 12 5 ]

The properties of these matrices are:

• Commutative property for addition  

This property informs that any two matrices with the same order can be added. Suppose the two matrices A and B are of the same order, then their commutative property of addition is the same, that is, A + B = B + A. 

[ 3 1] + [ 5 4] = [ 8 5 ] 

  [5 4 ]+[ 3 1 ]=[ 8 5 ]

• Associative property of addition 

Here if there are three matrices with the same order, then their position does not matter in addition. If there are three matrices A, B and C in a particular order, then according to this property: A +(B + C) = (A +B) + C

[ 1 7] + [ 3 8] + [ 2 5] = [ 4 15 ]+ [ 2 5] = [ 6 20 ]

• Additive identity property

As said before, zero matrices add to any matrix for the same results. According to this property, for matrix A exists a matrix O where A + O = A. 

[ 1 4 ]+ [ 0 0] = [ 1 4 ]    

 [ 5 3 ]+[ 0 0 ]= [5 3 ] 

• Additive inverse property

 In this there is a rule that the inverse of a matrix is -A of the same order. In other words, for matrix A, there is a unique matrix B. A + B = O 

[ 1 4] + [ -1 – 4 ]= [ 0 0 ]

[ 5 3 ]+[ – 5 – 3] =[0 0 ] 

• Closure property of addition

It where A+B=C, where C is a matrix with the same dimension as A and B

Conclusion

To sum up, matrix order is essential to define the equality of matrices. There are only a few matrices that can define the nature of matrices. Even if you use a rectangular matrix, the elements should be the same. In the equality of matrices, the two matrices should have the same order and corresponding elements. Matrices help us in our daily lives, and therefore, it is necessary to know about them. You can refer to study material notes on the equality of matrices to learn further. Matrices play a significant role in plotting graphs and many scientific disciplines. It represents real-life data such as population, mortality rate, etc.