B, you are Smaller than A

The kind of knowledge which is supported only by observations and is not yet proved must be carefully distinguished from the truth; it is gained by induction, as we usually say. Yet we have seen cases in which mere induction led to error. Therefore, we should take great care not to accept as true such properties of the numbers which we have discovered by observation and which are supported by induction alone. Indeed, we should use such a discovery as an opportunity to investigate more exactly the properties discovered and to prove or disprove them; in both cases we may learn something useful.

Leonhard Euler

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Bigger or smaller, as we all know, that entire mathematics is full of such questions. One gets an intuitive idea of bigger and smaller, probably at the age of around 5 years or so. 1 is smaller than 2 or 5 is bigger than 4 are appropriate for this stage. As we grow up, difficulty level increases and eventually we are asked to compare the numbers 100 and 101! With the passage of time, the turn comes to compare 23 and 32. This is the case when the student is studying in class 7.

When studying at the senior secondary level, a student now has to face the same kind of questions but with different inputs, more advanced than the earlier years. Of course, they study matrices in these classes. Yes, you are correct. In this article, we will consider an important example from matrices.

Consider two matrices A and B of the same order, say of order 3 × 3. In the following example, each corresponding element of B is smaller than that of A. So, is it possible to conclude B < A?

Here, A and B both are matrices of order 3. Also note that, each element of B is less than that of the corresponding element of A. for example, 1 < 20, 2 < 19, 5 < 10 etc.

As long as we are in the real domain, of course, exactly one of the three relations is true: either x > y or x = y or x < y. In other words, it’s easy to determine which one is smaller from two given real numbers.  Great, that’s enough to explain the given example.

Before we proceed to justify which one is smaller among A and B, let’s recall the definition of a matrix. We know that, a matrix is an arrangement of objects into rows and columns. From this definition, it is quite clear that the concept of smaller than is not possible to implement among the matrices. This just because, it does not have any real value! For example, consider 6 objects and let us arrange it as below. We can write, 6 = 1 × 6 = 6 × 1 = 2 × 3 = 3 × 2. Hence, all possibilities are

None of them have any real value. As a result, we cannot compare them and hence no concept of one is smaller than the other.

Perhaps, we can assign a real number (please excuse me for not considering the complex numbers) to every square matrix by means of its determinant. A determinant is a function between the set of all square matrices (of any finite order, say n) and real numbers. It follows that, every square matrix has a determinant value, which is a real number.

In the present scenario, after calculation we get det A = 1 and det B = 0. Clearly, 0 < 1 and so det B < det A. But, it is difficult to conclude that B < A!

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