Linear Algebra: Derivation of formula for the inverse of a matrix and Cramer’s rule

In this article I shall attempt to derive a formula for the inverse of a matrix, and from there derive Cramer’s rule. I feel that many textbooks and courses on linear algebra (especially at high-school level) present matrices and their corresponding formulas and definitions without giving any hint of where these ideas came from, and […]

A brief overview of ‘randomness’ and ‘noise’

What is pseudo-randomness? If you were asked to generate a number in your head in the bounds of 1 to 10, you may say that it’s a pretty easy task. You may even say that the probability of someone guessing your generated number is 1 in 10. However this is far from the case. People […]

Proving the Volume of a Sphere (Cylindrical Co-ordinates feat. triple integrals)

We are always taught that the volume of a sphere is 4/3πr^3 at school. But how can this be proved? One of the ways is to use cylindrical co-ordinates and integrate over suitable ranges in each of this co-ordinate system’s dimensions. But before we can do this, what are cylindrical co-ordinates? Cylindrical co-ordinates is similar […]

The Exponential Function – Why so useful?

An exponential function is a function where the input variable (usually written as x) is an exponent. Functions with constants in the exponent are also considered exponential functions. Then what is so amazing about these exponential functions? It all comes down to the rate of change of such a function (also known as the derivative […]