Matrices Multiplication
Now after we have Introduced Matrices and Explained Basic Matrices Operations in our previous blogging Matrix Algebra – An Introduction & Basic Operations we have introduced the following operations
- Matrices Constant Multiplication.
- Matrices Addition.
- Matrices Subtraction.
Today we will discuss the operation of Matrices Multiplication .
Let’s consider the following case : A company have two types of employees Permanent Employees and Contracted Employees we assume 40% of contracted Employees move to Permanent employees and 10% of Permanent employees moves to be contracted employees. Assume C be the contracted employees and P be Permanent employees. So after one yes the Permanent Employees population will be
0.9 P + 0.4 C
and the contracted employees population will be
0.6 C + 0.1 P
After 2 years the Permanent employees count will be 0.9(0.9 P + 0.4 C) + 0.4 (0.1 P + 0.6 C)
And Contracted Employees will be 0.6 (0.6 C + 0.1 P) + 0.1 (0.9 P + 0.4 C)
Is there a more concise way to represent company population after certain years ?!
ok , we speak about Matrices so you know that i’ll say the Matrices will do the job , yes it’s.
Let us represent the two populations in one table (meaning a column object with two entries):
So after one year the table which gives the two populations is
If we consider the following rule (Product of two Matrices)
let’s apply this on the population increase as follow
and from the previous rule we can see the result
And after 2 years the population could be
and the result will be
Which is the requested result as stated before , so we can conclude that matrix’s representation can give us more simple and neat solution to such cases.
So let’s go deeper and explain how Matrices Multiplication is done , let’s consider
how the process is done , we multiply each row in first matrix into every column in second matrix i.e 1st row in 1st matrix into 1st column in second matrix (aP + bC) and this give us the first row in the result & 2nd row in the first Matrix into 1st Column in second Matrix (cP + dC) and this gives us the second row in the resultant Matrix
We can see that for the Matrix’s Multiplication we don’t have to have two matrices with the same size but we only need the number or columns of the first matrix = number of rows in the second matrix. like
And we must to know that the above condition is a Must of matrices multiplication and the following operation is impossible to be done
So we will take care when speaking about matrices multiplication , we can’t say multiplication of matrices A and B because A x B and B x A are not equal even A and B are equal in size.let’s take this example to validate this rule
then
and
so the rule is Matrices Multiplication is not commutative and the order of matrices in multiplication process is a must condition.
let’s see the following example and see what we conclude.
this means that Multiplication of Two non zero matrices may give a Zero matrix.
