For AI agents: use /llms.txt for the Nakafa content index.
Matrix multiplication is a fundamental operation in linear algebra. Unlike matrix addition or subtraction where elements are operated on directly, matrix multiplication has its own specific rules.
Two matrices, let's say matrix and matrix , can be multiplied () only if .
Suppose matrix has an order of (meaning rows and columns) and matrix has an order of (meaning rows and columns).
Since the number of columns in matrix () is equal to the number of rows in matrix (), then matrix and can be multiplied.
The result of the multiplication, let's call it matrix , will have an order of .
The element in matrix (i.e., the element in the -th row and -th column) is calculated by multiplying each element in the -th row of matrix by the corresponding element in the -th column of matrix , and then summing all these products.
Mathematically, if and , then the element of matrix is:
The notation (sigma) means summation.
In the formula above, we sum the products for all values of from to .
Let's look at a simple example to understand the process.
Suppose we have matrices and .
Matrix has an order of and matrix also has an order of . The number of columns in (which is ) is equal to the number of rows in (which is ), so we can multiply them. The result, , will have an order of .
The elements of matrix are calculated as follows:
Given two matrices:
Matrix has an order of and matrix has an order of .
The number of columns in matrix (which is ) is equal to the number of rows in matrix (which is ).
So, can be calculated and will result in a matrix of order .
Let's calculate :
So, the result of the matrix multiplication is:
Now, what about ?
Matrix has an order of and matrix has an order of .
The number of columns in matrix (which is ) is not equal to the number of rows in matrix (which is ).
Therefore, the multiplication is undefined. This illustrates one of the important properties of matrix multiplication.
Matrix multiplication has several important properties:
Generally Not Commutative:
This means . We have already seen an example above where is defined but is not. Even if both are defined, the results are not necessarily the same.
Associative:
If the matrix multiplications and are defined, then holds. This means the order of grouping the multiplication does not change the final result.
Distributive:
Matrix multiplication is distributive over matrix addition or subtraction:
Multiplication by Identity Matrix ():
If is a square matrix of order and is the identity matrix of order , then:
Multiplication by a Scalar ():
If is a scalar (real number), then:
Matrix multiplication is very useful in various fields, one of which is for managing data and calculating aggregate values.
Imagine a home industry produces three types of food: tempeh chips, banana chips, and potato chips.
These foods are marketed in three places: Place A, Place B, and Place C.
The number of chips (in jars) sold in each place is presented in matrix . The columns in matrix respectively represent Place A, Place B, and Place C, while the rows respectively represent tempeh chips, banana chips, and potato chips.
The first row () means jars of tempeh chips were sold in Place A, in Place B, and in Place C.
The price for each jar of chips (in rupiah) is stated in the column matrix below:
To determine the total revenue from each type of chip across all places, we can multiply matrix by matrix .
However, pay attention to the order of the matrices. Matrix has an order of and matrix has an order of . The number of columns in () is equal to the number of rows in (), so can be calculated and will result in a matrix of order .
Matrix will show the total revenue for each type of chip.
From matrix , we can see that the total revenue from the sale of tempeh chips is , banana chips , and potato chips .
If the question is "determine the revenue matrix for each place", then we need to arrange the price matrix differently or perform multiplication with the transpose of .
Suppose we want to find the total revenue in Place A, Place B, and Place C. We can use the price matrix as a row matrix and multiply it by matrix : .
Matrix has an order of and has an order of . The result will have an order of .
Thus, .
This means the total revenue from Place A is , from Place B is , and from Place C is .
The interpretation of the elements of the resulting matrix greatly depends on how the initial matrices are defined and how the multiplication is performed.
Given the following matrices:
Determine the matrices and .
Is ?
Calculating :
Matrix has an order of and has an order of . The result will have an order of .
Calculating :
Matrix has an order of and has an order of . The result will have an order of .
From the results above, it is clear that and .
So, . This is another example showing that matrix multiplication is generally not commutative.
This holds if all involved addition and multiplication operations are defined.
The identity matrix acts like the number in ordinary number multiplication.
