Why not to use cofactor expansion

Here is a list of all the pessimistic reasons for wanting to teach cofactor expansion for computing determinants. These reasons are framed from the perspective of someone with cynical motivations, such as wanting to make exams more difficult, induce errors, or maintain strict control over the course average.

1. Exam difficulty and time constraints

Perfect for "time sinks" on exams

Why it works

Cofactor expansion is slow and tedious, especially for $4 \times 4$ or larger matrices. It requires students to compute multiple smaller determinants recursively, each of which takes more time. A single cofactor expansion question can devour 20+ minutes of an exam.

Motive

It eats up exam time, forcing students to rush through the rest of the test. The problem appears "easy" at first glance, but the recursive nature leads to an exponential explosion of calculations.

Example

"Calculate the determinant of this $5 \times 5$ matrix using cofactor expansion"—it will take forever and students will waste valuable exam time.

2. Tedious and error-prone process

Guaranteed source of student mistakes

Why it works

Cofactor expansion requires careful attention to signs $(-1)^{i + j}$, submatrices, and recursive determinants, making it easy to mess up a single sign, forget a term, or miscalculate a minor determinant.

Motive

This is a "gotcha" method for penalizing students. If they forget even one negative sign or make a small arithmetic mistake, the entire determinant is wrong. This gives an easy opportunity for partial marks or flat deductions.

Example

"Use cofactor expansion on the second row to compute $\textrm{det}(A)$."—the student needs to keep track of signs for every element, and small slip-ups can result in entirely wrong answers.

Sign rule confusion

Why it works

Students have to keep track of the signs $(-1)^{i + j}$. for every term in the expansion. Since sign mistakes are easy to make, this guarantees confusion and frustration.

Motive

It's a built-in method to "trip up" even good students. The seemingly simple logic of the sign flip becomes more confusing as the matrix size increases. Students might miscount, lose track of which row or column they’re on, or incorrectly apply the rule.

Example

"Use cofactor expansion along the third row." Half the students will get the sign wrong because they miscount $i$ and $j$, as if this has any impact on students being competent engineers.

Recursive nature is chaos for students

Why it works

Each cofactor expansion on an $n \times n$ matrix requires students to compute $(n−1)^2$ determinants, which themselves require expansion. Students can easily lose track of which minor they are calculating and for which row or column.

Motive

It generates cognitive overload, forcing students to juggle multiple small determinants in their heads. Mistakes compound as they go deeper into the recursive tree of calculations.

Example

"Calculate $\textrm{det}(A)$ for this $4 \times 4$ matrix using cofactor expansion." By the time they are working with multiple $3 \times 3$ determinants, students forget where they started.

3. Increases exam grading power

Partial marks galore

Why it works

Since cofactor expansion involves many intermediate steps (calculating minors, tracking signs, multiplying, and summing), there are multiple places where instructors can award "partial marks."

Motive

It's a way to create more granular grading. If a student does 90% of the problem correctly but forgets a sign, you can justify giving them only 70% of the marks. This ensures students are incentivized to work perfectly, but also allows graders to be "tough but fair."

Example

Award 3/10 marks if the student gets the minors right but forgets the sign. Award 1/10 marks if the student sets up the cofactor formula but makes a mistake early on.

Highly subjective grading

Why it works

The multi-step nature of cofactor expansion means different students might approach it slightly differently (choosing different rows/columns to expand), but mistakes at intermediate steps can result in wildly different paths.

Motive

This gives the grader absolute power over how partial marks are awarded. Two students may do 90% of the work correctly, but if one makes an arithmetic mistake and the other forgets a sign, they might get different grades.

Example

If a student chooses the "hardest row" to expand, the instructor can justify giving them fewer marks than another student who chose the "easier row" (even if it wasn’t explicitly stated in the question).

4. Illusion of complexity is equated to perception of rigour

Makes the course look harder

Why it works

Cofactor expansion looks sophisticated. It involves recursive definitions, submatrices, and abstract mathematical notation, which gives the impression of mathematical "depth."

Motive

It's a way to make the course appear more "rigorous" than it actually is. From the student's perspective, the complexity seems high, even though it's just mechanical recursion.

Example

Students see $\textrm{det}(A) = \sum_{k = 1}^n (-1)^{i + j}A_{i,j}$ and feel overwhelmed. This makes the course feel "deep" and "difficult."

Builds the illusion of competence

Why it works

Memorizing the steps of cofactor expansion makes students feel like they are doing "serious math," even though they aren't learning efficient computational techniques.

Motive

It promotes the illusion that students are developing "mathematical maturity," but they aren't learning the modern, efficient algorithms (like LU decomposition) used in practice. This ensures students remain dependent on the instructor's guidance.

Example

Instead of teaching computationally efficient methods like LU decomposition, the instructor emphasizes cofactor expansion, giving students false confidence in a "mathy-looking" process.

5. Tradition and resistance to change

"This is how I learned it" defence

Why it works

Many instructors themselves were taught cofactor expansion as the "default" way to compute determinants. There’s a legacy of passing down this outdated method as a "rite of passage."

Motive

It maintains a cycle of suffering. Students have to endure the same unnecessary effort their instructors went through. There's an element of "If I had to do it, so should you."

Example

"This is how it's always been done."

Avoids teaching more advanced (and useful) methods

Why it works

LU decomposition is more efficient but harder to explain conceptually. Cofactor expansion, despite being inefficient, is easier to introduce since it's recursive and "obvious" for small $2 \times 2$ and $3 \times 3$ matrices.

Motive

It's easier to avoid teaching modern, practical methods like LU decomposition because they require more explanation (pivoting, row swaps, etc.). Cofactor expansion fits neatly into a simple, clean recursive definition.

Example

Avoid teaching computationally efficient methods. Stick to "clean" cofactor expansion because it’s "simpler to explain," even if students never use it in practice.

For a summary of the reasons:

Cofactor expansion is inefficient, error-prone, and obsolete for large matrices. Yet, if the instructor's goal is to increase student suffering, maintain control of the course average, and create a perception of "rigour," it is the perfect tool. It has the "illusion of depth," generates errors, and is perfectly suited for "time-sink" exam questions.

In reality, LU decomposition is far more efficient, stable, and conceptually valuable, but it requires more up-front explanation, making it less "convenient" for instructors trying to create simple exam questions.

Why teaching cofactor expansion might be useful

Cofactor expansion provides students with a conceptual foundation for understanding determinants, recursion, and the decomposition of matrices. It introduces key ideas like breaking larger problems into smaller sub-problems, which directly relates to recursion in computer science and divide-and-conquer algorithms. It also allows students to see how determinants relate to geometric concepts like area, volume, and orientation. The cofactor expansion process reinforces attention to detail, as students must track signs, submatrices, and arithmetic carefully—all valuable skills for engineering, computer science, and mathematics. Moreover, cofactor expansion serves as a natural entry point for inductive proofs and theoretical results in linear algebra, as many key determinant properties (e.g., row swaps flipping signs, $\textrm{det}(AB) = \textrm{det}(A) \textrm{det}(B)$ are easiest to prove using this method. Finally, cofactor expansion is historically significant, reflecting how early mathematicians approached determinant computation before modern computational techniques like LU decomposition were developed.

Why these justifications don't hold up

Despite its conceptual value, teaching cofactor expansion as a computational method is unjustified. The recursive definition is elegant, but for matrices larger than $3 \times 3$, it is exponentially inefficient, with a time complexity of $O(n!)$. compared to the $O(n^3)$ of modern algorithms like LU decomposition. Teaching it as a "practical method" misleads students into thinking it's useful for computation, even though no one in practice would compute a $10 \times 10$ determinant using cofactor expansion. The argument that it "trains attention to detail" is weak because many better tasks (like debugging code or conducting row-reduction) also train attention while offering real-world utility. While it provides an intuitive introduction to recursion, the time spent on cofactor expansion could be better used teaching recursion in contexts where recursion is actually applied (like depth-first search or divide-and-conquer algorithms). Historical significance is also a shallow justification—we don't teach outdated methods like Newton's fluxions in calculus classes. Ultimately, students gain far more from learning computationally efficient methods (like LU decomposition) or conceptually rich approaches (like eigenvalues and linear transformations) than from suffering through a laborious, error-prone, and obsolete method like cofactor expansion.