Sungjai's Recreational Mathematics

Matrices are usually first introduced to students in pre-calculus and only the most basic operations and applications are taught to them. For example, we are taught how to add, multiply a scalar value, multiply matrices, and how to find their Inverses. We are taught that only the square matrices have inverses, but is this really true? Today, we will explore this concept of the Inverse of Non-Square Matrix. The Inverse of Matrix Product First, let us explore how the Inverse of a Product relates to Inverses of its Factors. Consider this matrix product: $$AB = C$$ where $A$, $B$, and $C$ are all matrices of legal orders. Now let us ask, how does $A^{-1}$ and $B^{-1}$ relate to $C^{-1}$? Let's define what we mean by Inverse of $C$. We want the inverse to have the property that $$CC^{-1} = C^{-1}C = \textbf{1}$$ where $\textbf{1}$ is a identity matrix. First, let's look at the first case: $$CC^{-1} = \textbf{1}$$ We can substitute the definition of $C$ and find that $$(AB...

Simpson's Rule, which is a extension of Trapezoidal Rule, which itself is a extension of Riemann Sum, is a approximation of true integral using second degree polynomial (rather than first degree polynomial of Trapezoidal and zeroth degree of Riemann Sum). Simpson uses the fact that $$p(x) = Ax^2 + Bx + C, \\ \int_{a}^b p(x) dx = \frac{b-a}{6}(p(a)+4p(\frac{a+b}{2})+p(b))$$ and extends to approximate that $$\int_{a}^bf(x) dx \approx \frac{b-a}{3n}[f(x_0)+4f(x_1)+2f(x_2)+4f(x_3)+\dots +4f(x_{n-1})+f(x_n)]$$ Now then, surely if second order polynomial yielded a closer approximation than first, then third order polynomial would be better approximation than second as well. What does it mean to use Third-degree Polynomial? For zeroth degree polynomial, we pick a single point in the interval to represent the curve. We cannot know about the slope or any concavity with a single point, and so a zeroth degree polynomial (a constant) is needed. This is what Riemann Sum uses to approx...

Last time, we covered Geometric Series and before that we covered Arithmetic Series. Now we will try to find formulas for more unusual Series such as $\sum\sin$ and $\sum\cos$. $$\sum_{i=n}^m \sin(i) = \int_{n-1}^mȢ\{\sin(x)\}dx, \\ \sum_{i=n}^m \cos(i) = \int_{n-1}^mȢ\{\cos(x)\}dx $$ Essence Identity $$Ȣ\{f(x+1) - f(x)\} = \frac{d}{dx}f(x+1) $$ Last time we were able to calculate Essence of Exponentials with help of this Identity (also called Essence Formula). This will also help us calculate Essence for Sine and Cosine, though through a more complicated process. Unlike Exponentials, the difference cannot be grouped into a single term. To compute $Ȣ\{\sin(x)\}$ and $Ȣ\{\cos(x)\}$, we will have to solve them simultaneously side-by-side. $Ȣ\{\sin(x)\} \text{ and }Ȣ\{\cos(x)\}$ Firstly, let us use Essence Identity to find that $$\begin{align*} Ȣ\{\sin(x+1)-\sin(x)\}&=\frac{d}{dx}\sin(x+1)\\ Ȣ\{\sin(1)\cos(x)+\cos(1)\sin(x) - \sin(x)\}&=\cos(x+1)\\ Ȣ\{\sin(1)\cos(x) + ...