Matrix Multiplication

Matrix multiplication is a binary operation that produces a matrix as its result. It is not similar to arithmetic multiplication because it is not commutative, which means the order of matrices matters. For instance, multiplying matrix A by matrix B (AB) does not give the same result as BA (AB ≠ BA).  
Two matrices are considered compatible when the number of columns in the first matrix is equal to the number of rows in the second. If matrix A has dimensions m × n and matrix B  has dimensions n × p, then they can be multiplied. If we multiply matrix A and matrix B, the result is matrix C of the order m × p. Take a look at the given image to gain a clearer understanding of the concept.
 

Matrix multiplication is different from arithmetic multiplication, so the former follows certain rules and properties in linear algebra. Here are the key properties of matrix multiplication:

Non-commutative: If we multiply two matrices A and B, the order matters, which means AB is not equal to BA.    
                         AB ≠ BA
 

Distributivity: The multiplication of matrices is distributive, and all the matrices in the multiplication are compatible.           
            A (B + C) = AB + AC 

Product with scalar: If c is a scalar (any number) and the product AB is defined, then:
      c (AB) = (cA) B = A(cB)
This property states that multiplying a scalar with either matrix, before or after multiplication, without changing the result.

Transpose: According to this property, 
                         (AB)T = BT AT 
             Here, T is the transpose. Hence, the transpose of the product of matrix A and matrix B is equal to the product of their transposes in reverse order. 

Complex conjugate: If the matrices have complex numbers, the conjugate of their product equals the product of their conjugates in reverse order.
        (AB)* = B*A*

Associativity: If we multiply three matrices A, B, and C, the grouping of the matrices does not affect the final result.       
  (AB)C = A(BC) 

Hence, the matrix multiplication is associative. 

Determinant: The product of the determinants is equal to the determinant of a product. 
       det (AB) = det A * det B
 

Matrix multiplication is the process of multiplying each element in a row of matrix A by each element in the corresponding column of matrix B, then adding the results to obtain the new matrix. We must follow several steps in order to multiply matrices:

Step 1: Before multiplying two matrices,check whether the number of rows in the second matrix equals the number of columns in the first matrix. 

Step 2: To begin, multiply the first element in the row of the matrix A with the corresponding element in the column of the matrix B. Next, continue the multiplication of matching elements and then add all these products together.  

Step 3: Write the added products in the correct positions of the new matrix. 

To understand the steps, let us look at an example.  
Multiply the given matrices. 

Matrix A: 

(Matrix A = 3 × 2)

Matrix B: 

(Matrix B = 2 × 1)

In this instance, the number of columns in matrix A and the number of rows in matrix B are equal. Therefore, the product matrix is a 3 × 1 matrix with 3 rows and 1 column. 
Now, multiply the given matrices: 

     
Next, we can write the results in the respective positions of the new matrix. 
AB = 

    
Therefore, the product matrix is:

When doing matrix multiplication, it is important to follow several rules. The compatibility of the two matrices is the primary requirement. It indicates the number of columns in the first matrix and the number of rows in the second matrix must be equal. 
If the order of the matrix A = m × n 
The order of the matrix B = n × p 
The order of product matrix = m × p 

For instance, if matrix A = 2 × 4 and matrix B = 4 × 2, the product is 2 × 2.

The process of multiplying 2 × 2 matrices follows the same principle as multiplying matrices of any order. Each element in a row of matrix A is multiplied by the corresponding element in a column of matrix B, and the results are summed. 

For example, matrix A =
 

Matrix B = 

Now, we can multiply element by element: 

The product (AB) is: 

The matrix multiplication formula can be used to multiply two compatible 3 × 3 matrices. A 3 × 3 matrix will also be the result of this multiplication. The equation is: 
 

For example, let us multiply two 3 × 3 matrices. 

Matrix A = 

Matrix B = 

Now, let us start the multiplication. 

Therefore, the product matrix is: 
AB = 

In technology, science, and mathematics, matrix multiplication plays an important role which helps to solve complex problems. Here are some real-life applications of matrix multiplication: 

• In computer design and animation, the developers use matrix multiplication to rotate, scale, or convert 2D and 3D objects in video games. For example, to change the angle of a character’s feet in a video game, matrix multiplication is useful. 

• In engineering and physics, professionals use the multiplication of matrices to solve complex equations of systems and to simulate physical systems. For instance, engineers use matrix multiplication to model electrical circuits and predict current flow and voltage distribution. 

• In sound processing and audio editing fields, sound engineers use matrix multiplication to compress audio files and to reduce the noise in audio. For example, professionals use this multiplication to adjust the pitch of various audio files which ensures high quality.