#1 Determinants & Where to Find Them

Hello! I hope y’all are having a great day. A while back when my friends and I were preparing for winter exams, one of them asked me “Hey, Ivanna — what is a determinant?” I thought about that question for a bit and realized that I had no clue. I knew it had something to do with matrices, but I had completely forgotten anything else. I replied with an Idk and got to searching on the web. After some research, I realized that high school usually only mechanically teach students how to find determinants. They don’t teach them what they actually mean. I then realized that determinants have actually popped up in other places over my past year of math studying. So…I went on a mini deep dive and made this article to answer my friend’s question and to share some uses for the determinant.

What is a determinant?

Since a determinant is a value related to a square matrix, let’s start from how to define a square matrix. Any square matrix can be used to transform a space from one coordinate plane to another. To denote the amount of dimensions (variables), the matrix will be represented as $n$ x $n$ ( $n$ -rows and $n$ -columns) where n is the dimension amount. So a 2×2 matrix will transform a 2D space, a 3×3 matrix will transform a 3D space and so on. At least for now we will assume that a matrix can only transform linearly, meaning that it can stretch, squash and rotate the space uniformly.

We call it linear because of this uniform change (when the change is not uniform, we can’t represent it as a matrix. We will get into why in the Jacobian section). The University of Texas hosts a fun simulation to play around with. This is all well and good, but what is a determinant?? Well a determinant is the coefficient by which space shrinks or grows after the space was transformed. For example the matrix $\begin{bmatrix}2 & 0 \\ 0 & 2 \end{bmatrix}$ grows a unit square’s area from $u^2$ to $4u^2$ ; 4 is the determinant of the matrix. An interesting thing to note: a determinant isn’t necessarily positive. It can be negative if the transformation has “flipped” the plane or even 0 if the transformation shrunk the space to a lower plane (we will discuss what this means in the parallelogram and cross product section). Now that we conceptually know what a determinant is, we can learn to calculate it.

How to calculate the determinant?

Calculating a determinant becomes easy once you know a few things: First, the determinant of a 1×1 matrix is the matrix’s element; and second, the minor of an element is the matrix of the left over elements after getting rid of the chosen element’s rows and columns. To calculate the determinant all we do is find the first row’s elements’ minor’s determinants; we then find the minor’s determinants the same way ‘till we get to 1×1 minors. Then we add the odd columns’ products and subtract the even columns’ products. That all sounds complicated and wordy (it definitely is wordy), but an example might expedite things. Let’s take the 3×3 matrix $\begin{bmatrix}3 & 2 & 3 \\4 & 5 & 6 \\7 & 8 & 9\end{bmatrix}$ . To find $\begin{vmatrix}3 & 2 & 3 \\4 & 5 & 6 \\7 & 8 & 9\end{vmatrix}$ ( $\begin{vmatrix}A\end{vmatrix}$ is the determinant while $\begin{bmatrix}A\end{bmatrix}$ is the matrix) we multiply the top elements by their minor’s determinant:

$3\cdot \begin{vmatrix}5 & 6 \\8 & 9\end{vmatrix}, 2\cdot \begin{vmatrix}4 & 6 \\7 & 9\end{vmatrix} , 3\cdot \begin{vmatrix}4 & 5 \\7 & 8\end{vmatrix}$

Then we find $\begin{vmatrix}5 & 6 \\8 & 9\end{vmatrix}$ by multiplying the top element by its minor’s determinant:

$5\cdot \begin{vmatrix}9\end{vmatrix} , 6\cdot \begin{vmatrix}8\end{vmatrix}$

If we recall our rule about 1×1 matrices we find that this minor’s first product is 45 and the other is 48. We now add the odd column’s products and subtract the even column’s products: 45 – 48. Our minor’s determinant is then -3.(I’ll only guide you through finding the first minor’s determinant because the rest use the same process) Now that we found our minor’s determinant (-3,-6,-3) we can multiply them by the their respective first row elements (getting -9, -12, -9). Now we again add the odd columns’ products and subtract the even columns’ products (- 9 + 12 – 9). This procedure leaves us with a determinant of -6 for our matrix $\begin{bmatrix}3 & 2 & 3 \\4 & 5 & 6 \\7 & 8 & 9\end{bmatrix}$ . Now that we know how to calculate the determinant, let’s get into some of my favorite uses for determinants.

#1 Solving linear equations(Using Cramer’s Rule)

One of the coolest uses for determinants is having an explicit formula for solving linear equations. While multiple people in the 18th century published the same formula or rule, the name of Cramer’s Rule is attributed to Gabriel Cramer, a Genevan mathematician. I won’t prove the formula here but if you want to see how to prove it both algebraically and geometrically here is a great Wikipedia article. The formula itself uses determinants to find each variable. For example, let’s say we are given 3 equations with 3 variables:

$\begin{array}{rcl}a_{1}x + b_{1}y + c_{1}z & = & k_{1} \\a_{2}x + b_{2}y + c_{2}z & = & k_{2} \\a_{3}x + b_{3}y + c_{3}z & = & k_{3}\end{array}$

To find any of the variables ( $x$ , $y$ , or $z$ ), we will find the determinant D of the coefficients:

$\begin{vmatrix}a_{1} & b_{1} & c_{1} \\a_{2} & b_{2} & c_{2} \\a_{3} & b_{3} & c_{3}\end{vmatrix}$

Next, we find the determinant D $_{x}$ for variable $x$ by replacing $x$ ’s coefficients with the constants:

$\begin{vmatrix}k_{1} & b_{1} & c_{1} \\k_{2} & b_{2} & c_{2} \\k_{3} & b_{3} & c_{3}\end{vmatrix}$

Finally, $x$ = D $_{x}$ /D. You can repeat this process for variables $y$ and $z$ (Side note: if the coefficient determinant is 0, then there is no single solution; the set can have no or infinite solutions). We can do this for any amount of unknowns, given we have as many equations that are linear. This method greatly simplifies solving for a system of linear equations — as long as they have a unique solution.

#2 parallelogram area and the cross product

Haha! I tricked y’all! This is a section covering not one…but two concepts! I paired these because of their relevance to each other. Some of you may know various ways to solve for the area of a parallelogram:

$side _{1}\cdot side _{2}\cdot\sin(\Theta)$

There is also another way that involves determinants! This method represents the parallelogram sides as vectors and uses them in a 2×2 matrix.

$\begin{bmatrix}4 & 1 \\0 & 2\end{bmatrix}$

As you may have guessed, the determinant of this matrix is the area of the parallelogram.

$\begin{vmatrix}4 & 1 \\0 & 2\end{vmatrix} & = & 8$

This method works because we treat the parallelogram as a linearly transformed unit square, where the transformation of the y axis is one vector and of the x axis is the other vector. The method works more efficiently in higher dimensions when calculating the volume of a parallelepiped or higher dimension parallelogram. Now that we’ve covered our basis: what is a cross product, and why is it relevant here? A cross product of two 3D vectors is the vector that is perpendicular to two other vectors (this works for higher dimensions too if you add another vector). Furthermore, the cross product can be used to find the formula for a plane that holds the two vectors.

But here is where it gets interesting: to calculate the cross product, all you do is find the volume (or the space in higher dimensions) of a parallelepiped (or the equivalent for higher dimensions) where the sides are the first vector and second vector. A third new vector (which will become our cross product vector) is an unknown vector with the variables $i$ , $j$ , $k$ (vector talk for $x$ , $y$ , $z$ ):

$\left\langle a_{1}, b_{1}, c_{1} \right\rangle\times \left\langle a_{2}, b_{2}, c_{2} \right\rangle & = & \begin{vmatrix}i & a_{1} & a_{2} \\j & b_{1} & b_{2} \\k & c_{1} & c_{2}\end{vmatrix}$

For easier calculation, most articles and textbooks show the transpose of this matrix because the determinants of a transpose and its matrix are the same. I specifically wanted to show this matrix to show the cross product’s relationship to the parallelogram’s area. Some nice uses for the normal vector is in Newtonian physics. Ever noticed the normal force or how torque is perpendicular to the force and radius vectors? Even the motion of light/electromagnetic waves when interacting with obstacles uses normal vectors.

#3 Jacobian Matrix

Up until now, we have only been talking about linear transformations. While we’ve discussed a matrix definition, parallelogram area and cross products, we have not touched on non-linear transformations. So what is a non-linear transformation? A non-linear transformation is distinct from a linear transformation because it lacks a constant change in space between the original coordinate plane and the new coordinate plane. A linear transformation is like a straight line with a constant rate of change (like a constant change in space or determinant), while a non linear transformation is like a function like a parabola with a varying rate of change (like a varying change in space). So if we can represent a linear transformation using a matrix, can we represent a nonlinear transformation using a matrix? Short answer, no; long answer, no…kinda but not entirely. To understand why, we need to know some calculus and partial derivatives. I won’t explain the basics of calculus because that would make this section too long, but here is a great article describing the basics of partial derivatives. The partial derivatives will help us understand how a linear transformation goes from a function to a matrix. Let’s start by taking the linear transformation from $x_{i}$ , $y_{i}$ coordinates to $x_{f}$ , $y_{f}$ coordinates where:

$\begin{array}{rcl}x_{f} & = & ax_{i} + by_{i} \\ y_{f} & = & cx_{i} + dy_{i}\end{array}$

If we were to take the partial derivatives of the initial coordinates with respect to the final coordinates, we get:

$\begin{array}{rcl}\partial x_{f}/\partial x_{i} &=& a \\ \partial x_{f}/\partial y_{i} &= &b \\ \partial y_{f}/\partial x_{i}& = &c \\ \partial y_{f}/\partial y_{i} &=& d\end{array}$

Notice that each element in a matrix (representing a linear transformation) is a number with no variables. If we put each partial derivative into a 2×2 matrix — where each column is an initial coord and each row is a final coord — we get $\begin{bmatrix}a & b \\c & d\end{bmatrix}$ . If we multiply the matrix by $\begin{bmatrix}x_{i} \\y_{i} \end{bmatrix}$ we get the original linear transformation function! We can also find partial derivatives of nonlinear functions and represent them in a matrix:

$\begin{bmatrix}\partial x_{f}/\partial x_{i} & \partial x_{f}/\partial y_{i} \\\partial y_{f}/\partial x_{i} & y_{f}/\partial y_{i}\end{bmatrix}$

This type of transformation matrix is called the Jacobian Matrix! (oh my god she said the name of the section!) Unfortunately, we can’t get the original functions from multiplying the Jacobian Matrix by the initial variables, so the Jacobian Matrix cannot be used to transform points. But the Jacobian Matrix has another use; it can tell us the change in space at the point that is being transformed. How? Determinants! Yeah, that’s right. When you thought you could escape, they came back. A Jacobian Matrix’s determinant (named the Jacobian) can show how a nonlinear transformation changes the space at a certain point. Because of its close relationship to derivatives (which are approximations of the slope of a tangent line to a function), the Jacobian Matrix plays a similar role with transformations in the sense that the Jacobian matrix linearly approximates the change in space. In real life, the Jacobian Matrix is used in machine learning, statistics, and some areas of physics like fluid mechanics.

Honorable Mentions

I didn’t include these concepts in my list as uses of determinants because they don’t primarily use the determinant. That said, I personally think these are some cool mathematical concepts. Firstly, eigenvectors and eigenvalues (and other ‘eigen’ stuff) demystify the complexities of linear transformations because they only scale instead of change their position. The whole ‘eigen’ stuff in math generally means that it can be operated on by a certain operator(whether that be a normal equation or something like a derivative) and only change by scale. Eigenfunctions for example come in handy in quantum mechanics, calculating the stationary states (states that don’t change over time) of quantum particles (like electron orbitals). While I can’t do it justice here, this video by Jack Star does a great job introducing eigenvectors and eigenvalues. Furthermore, the covariance matrix describes the variance in multivariate statistics. Covariance matrices are used in Principle Component Analysis to clarify a set of multivariable values. Here is a wonderful article on medium that discusses PCA. This is in no way exhaustive; the list is only meant to take a glimpse at the determinant’s wide span in math.

To conclude, indubitably

Hopefully, this article was able to shed a bit of light on other aspects in math. I hope I was able to answer my friend’s question, even if I was two months too late. Disclaimer: I’m only a highschooler, so feel free to correct me if I get something wrong. Do tell me what I should look into next — and have a good day!

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