Intro
A clever way to protect the private information we send to each other is through encryption and decryption processes involving inverse matrices.
To ensure that an eavesdropper cannot read a message sent electronically, we encrypt it, scrambling and rearranging the symbols to look like nonsense.
In order for the right person to read it, the message must then be decrypted upon receipt.
Using a technique called "Hill Cipher", the sender and receiver agree on a matrix to use to encrypt messages. Since the receiver knows that, they can use the inverse of the matrix to decrypt the message. This makes the message readable again.
This is a well-known solution in cyber security, and is often used as one of the steps in reliable cryptographic algorithms.
Concept
Multiplying any non-zero number by gives you back the same number. This also means that dividing a number with itself gives you .
Another way of saying that we divide by a number , is to say that we multiply by its multiplicative inverse .
The inverse to a matrix is another matrix, and the product of the matrix and its inverse is the identity matrix.
In fact, there is no such thing as division, it is just a term used for multiplying by an inverse.
We can, however, multiply by the inverse of a matrix , denoted as , resulting in the identity matrix , which we can think of as the equivalent of the number for matrix multiplication.
Math
There is no guarantee that exists. For instance, must be a square matrix.
If the inverse of a matrix does exist though, then the following fact makes the inverse of a matrix a powerful tool.
Consider the familiar equation:
Just the existence of tells us that there is a unique solution. Furthermore, we find the vector relatively easy if we multiply both sides of the equation by :
Hence, by carrying out the multiplication will automatically generate the vector that is our desired solution.
Inverse matrices
Motivation
Let be an -matrix. If there is another -matrix such that the matrix multiplication of them results in the identity matrix , this matrix is called the inverse and is called . Then it is important that:
We will now justify the existence of the inverse of . To do that, we need to start from elementary matrices. Briefly, each individual elementary row operation can be expressed as a matrix multiplication, which is then called an elementary matrix.
Here are some examples:
whereupon multiplied by an equal-dimensional matrix
multiplies the second row by -3.
changes places on lines 1 and 4.
sums the first line with five times of the third line.
So, since each row operation can be expressed as an elementary matrix , it means that if an -matrix can be row reduced to using row operations, is expressed as:
From above we can define:
To be completely satisfied, we also want to be able to justify that is invertible. We rely on the fact that each elementary matrix is invertible because it performs a single row operation on each matrix and partly because each row operation is invertible.
Let be the inverted operation of , then we have:
This means that we can invert the above definition of the inverse matrix because it follows that:
This is the basis of a beautiful proof of the existence of the inverse , if and only if, can be row reduced to . Then also can be inverted which leads back to .
From the three cases for each linear system of equations, we know that when the row-reduced matrix becomes , the solution set is a single point, i.e. a unique solution. Hence this theorem follows:
For each -matrix , the following statements are either completely true or false.
The reduced row echelon form of is .
can be expressed as the product of elementary matrices.
is invertible, i.e. exists.
has a unique solution for every .
When an inverse to exists, we see the strength in the solution of a system of equations, because we have:
Find the inverse
To find the inverse , we assume that it can, if it exists, be expressed as the product of k elementary matrices in an equation where the right-hand side is . In other words:
is the unknown for which we want to solve the equation above, which is like in the equation . If we convert the above to an augmented matrix, we therefore get:
If the above equation system has solutions, it will have exactly one solution, namely the sought after inverse . To find out if the system has a solution, we use the Gaussian-Jordan method.
Example
Let:
We set up an augmented matrix for the above and get:
With the Gaussian-Jordan method, we get:
Now we have a left term in the augmented matrix as exactly , which means that the system has a unique solution, whereupon the right term is exactly that we solved for.
In summary, we have:
Inverse of 2x2 matrices
For -matrices, there is a formula for finding the inverse if it exists. It is connected to the determinant.
If the determinant has not yet been introduced in this syllabus, you can turn that thirst for knowledge upside down and focus on the first line in the definition below.
Let:
then it applies to the inverse that:
