Math - Page 7

We’re continuing to prepare math tutorials. Hopefully, it will be helpful for you and you won’t be in need asking our experts something like “Do my math homework for me, please! I’m stuck..” This is the second part of our tutorial explaining how to calculate determinants. We’re asked to calculate the determinant of  the following 4×4 matrix:

A=\begin{pmatrix}-1 & {1} & 4 & 2 \\2 & -1 & 2 & 5 \\1  &2  & 3& 4\\3& 4& -1 & 2\end{pmatrix}

Recall that only square matrices have a determinant, for non-square ones it’s not defined. Matrix A is a square 4×4 matrix so it has determinant. Here we have no zero entries, so, actually, it doesn’t matter what row or column to pick to perform so called Laplace expansion. Let it be the first column. It means that we set j=1 in general formula for calculating determinants which works for determinants of any size Continue reading →

We’re continuing to consider Gaussian elimination method. Earlier we’ve prepared several tutorials covering theoretical background and examples including application of matrix representation. Yet, all those examples suggested systems of linear equations which have a solution. That’s not always so, though. In your homework or linear algebra assignment you can face different situations. In fact, there are three possibilities: system of linear equations can have either unique solution, or infinitely many solutions, or no solution at all. We’ll discuss the last case in this section.

Let’s solve the following system of linear algebraic equations:

\left\{ \begin{aligned}2x_1-x_2+3x_3=4\\-3x_1+2x_2+x_3=5\\-4x_1+2x_2-6x_3=1 \end{aligned}\right. Continue reading →

In this section we’re going to discuss the most important properties of parabola, also we’ll show how to find its vertex, roots and intercepts. This information is needed to sketch the graph and also allows to analyze the behavior of given parabola. We’ve already shown how to sketch graphs of quadratic function, and now we’re going through most essential details. Let’s start with considering  the equation of parabola. Continue reading →

What is a parabola? How to sketch its graph? These are the questions for this section. Sketching parabolas is commonly asked in algebra homework tasks, so some tips will be helpful. We’ll discuss the basic concepts and show you on simple examples how to deal with parabolas.

Let’s get started. In simple words, parabola is a U-shaped curve with some specific properties. However, note that not any U-shaped curve can be called parabola. It’s a common fact that math likes to describe everything by formulas. No surprises here, we also can write a formula for parabola. Basically, any parabola is described by some quadratic equation.  Continue reading →

In previous sections we’ve talked about Gauss elimination method and considered some examples. Today we have one more systems of linear equations to solve, but this time we’ll apply matrix representation for systems of linear equations:

\left\{\begin{aligned}2x_1+4x_2-3x_3+2x_4=0\\x_1-2x_2+5x_3+2x_4=2\\2x_2-4x_3+5x_4=-1\\-2x_3+x_4=-3 \end{aligned}\right. Continue reading →

If you need to solve a system of linear equations, Gaussian elimination is often the best choice. We’ve considered this topic in one of the previous sections. For better understanding we prepared several examples covering this topic similar to those in linear algebra homework. Now let’s solve the system of 3 linear equations with 3 unknowns x,y,z Continue reading →