International Standards for Matrix Algebra

Matrices, though they may appear weird objects at first, are a very important tool in expressing and discussing problems which arise from real life cases.
Our first example deals with economics. Indeed, consider two families A and B (though we may easily take more than two). Every month, the two families have expenses such as: utilities, health, entertainment, food, etc... Let us restrict ourselves to: food, utilities, and health. How would one represent the data collected? Many ways are available but one of them has an advantage of combining the data so that it is easy to manipulate them. Indeed, we will write the data as follows:

If we have no problem confusing the names and what the expenses are, then we may write

This is what we call a Matrix.”

Before we give the formal definition of how to multiply two matrices, we will discuss an example from a real life situation. Consider a city with two kinds of population: the inner city population and the suburb population. We assume that every year 40% of the inner city population moves to the suburbs, while 30% of the suburb population moves to the inner part of the city.

Let I_o be the initial population of the inner city and S_o be the initial population of the suburban area. So after one year, the population of the inner part is I_1=0.6 I_o+0.3S_o , while the population of the suburbs is S_1=0.4I_o+0.7S_o . After two years, the population of the inner city is I_2 =0.6 (0.6 I_o + 0.3 S_o) + 0.3 (0.4 I_o + 0.7 S_o) and the suburban population is given by S_2 =0.4 (0.6 I_o + 0.3 S_o) + 0.7(0.4 I_o + 0.7 S_o) . Is there a nice way of representing the two populations after a certain number of years? Let us show how matrices may be helpful to answer this question. Let us put the coefficients in one matrix and the two populations in a table (meaning a column object with two entries): the coefficients: \left( {\matrix{ {0.6} & {0.3} \cr {0.4} & {0.7} \cr } } \right) and the populations: \left( {\matrix{ {I_o } \cr {S_o } \cr } } \right) . Then, \left( {\matrix{ {I_1 } \cr {S_1 } \cr } } \right) = \left( {\matrix{ {0.6I_o + 0.3S_o } \cr {0.4I_o + 0.7S_o } \cr } } \right) .

If we consider the following rule for matrix multiplication: \left( {\matrix{ a & b \cr c & d \cr } } \right) \cdot \left( {\matrix{ I \cr S \cr } } \right) = \left( {\matrix{ {aI + bS} \cr {cI + dS} \cr } } \right) ,

we have: \left( {\matrix{ {I_1 } \cr {S_1 } \cr } } \right) = \left( {\matrix{ {0.6} & {0.3} \cr {0.4} & {0.7} \cr } } \right) \cdot \left( {\matrix{ {I_o } \cr {S_o } \cr } } \right)   and   \left( {\matrix{ {I_2 } \cr {S_2 } \cr } } \right) = \left( {\matrix{ {0.6} & {0.3} \cr {0.4} & {0.7} \cr } } \right) \cdot \left( {\matrix{ {0.6} & {0.3} \cr {0.4} & {0.7} \cr } } \right) \cdot \left( {\matrix{ {I_o } \cr {S_o } \cr } } \right) .

There are many ways to encrypt a message. And the use of coding has become particularly significant in recent years (due to the explosion of the internet for example). One way to encrypt or code a message uses matrices and their inverse. Indeed, consider a fixed invertible matrix A. Convert the message into a matrix B such that AB is possible to perform. Send the message generated by AB. At the other end, they will need to know A^{-1} in order to decrypt or decode the message sent. Indeed, we have: A^{-1}(AB)=B , which is the original message.

Keep in mind that whenever an undesired intruder finds A, we must be able to change it. So we should have a mechanical way of generating simple matrices A which are invertible and have simple inverse matrices. Note that, in general, the inverse of a matrix involves fractions which are not easy to send in an electronic form. The best is to have both A and its inverse with integers as their entries. In fact, we can use our previous knowledge to generate such class of matrices. Indeed, if A is a matrix such that its determinant is +1 or -1 and all its entries are integers, then A-1 will have entries which are integers.

So how do we generate such class of matrices? One practical way is to start with an upper triangular matrix with +1 or -1 on the diagonal and integer-entries. Then we use the elementary row operations to change the matrix while keeping the determinant unchanged. Do not multiply rows with non-integers while doing elementary row operations. Let us illustrate this on an example.

Example. Consider the matrix \left( {\matrix{ { - 1} & 5 & { - 1} \cr 0 & 1 & 8 \cr 0 & 0 & 1 \cr } } \right) .

First we keep the first row and add it to the second as well as to the third rows. We obtain \left( {\matrix{ { - 1} & 5 & { - 1} \cr { - 1} & 6 & 7 \cr { - 1} & 5 & 0 \cr } } \right) .

Next we keep the first row again, we add the second to the third, and finally add the last one to the first multiplied by -2. We obtain \left( {\matrix{ { - 1} & 5 & { - 1} \cr { - 2} & {11} & 7 \cr 1 & { - 5} & 2 \cr } } \right)

This is our matrix A. Easy calculations will give det(A) = -1, which we knew since the above elementary operations did not change the determinant from the original triangular matrix which obviously has -1 as its determinant. We leave the details of the calculations to the reader. The inverse of A is \left( {\matrix{ {57} & { - 5} & {46} \cr {11} & { - 1} & 9 \cr { - 1} & 0 & { - 1} \cr } } \right)

Back to our original problem. Consider the message “I LOVE MONICA”

To every letter we will associate a number. The easiest way to do that is to associate 0 to a blank or space, 1 to A, 2 to B, etc...

Another way is to associate 0 to a blank or space, 1 to A, -1 to B, 2 to C, -2 to D, etc...

Let us use the second choice. So our message is given by the string

Now we rearrange these numbers into a matrix B (with 3 rows so that we can multiply AB). For example, we have B = \left( {\matrix{ 5 & 8 & 0 & { - 7} & 1 \cr 0 & { - 11} & 7 & 5 & 0 \cr { - 6} & 3 & 8 & 2 & 0 \cr } } \right)

Then we perform the product AB, where A is the matrix found above. We get

The encrypted message to be sent is: 1, -52, -7, -66, …, -1, -2, 1