Laplace Expansion and Formal Definition of Determinants

 

Introduction

We’ve already seen how to calculate determinants of 2×2 and 3×3 matrices. Now let’s take it a step further: 4×4 matrices. Here, the most commonly used method is Laplace expansion. We’ll also present the formal definition of a determinant using permutations — a bit more abstract, but essential to understanding how everything fits together.

Determinant of a 4 × 4 Matrix: Laplace Expansion

Consider the matrix:

Choose a row or a column (usually the first row). For each element in the row, eliminate its row and column, and compute the determinant of the remaining 3×3 submatrix. The determinant will be:

                                        det(M) = a × A₁₁ − b × A₁₂ + c × A₁₃ − d × A₁₄

where Aᵢⱼ is the determinant of the submatrix obtained by removing row i and column j, and the signs must alternate: +, −, +, −.

A 4×4 matrix thus has determinant:

Example

Choosing the 1st row:

🔹 Eliminate the row and column of each element in the 1st row

🔹 Compute the determinants of the remaining 3×3 matrices

🔹 Apply the signs: + − + −

We choose the first row because it contains a zero, simplifying the calculations:

For a₁₁, the determinant is:

                        

For a₁₃, the determinant is:

          

For a₁₄, the determinant is:

The determinant of M is:                                                  

Formal Definition of a Determinant (Any Order)

The determinant of a square matrix of order n is given by the sum of n! products, each formed by taking one element from each row and each column, multiplied by (−1)^p, where p is the parity of the permutation of the column indices.

Important Theorem

Each term in the determinant expansion contains exactly one element from each row and each column. This ensures that the determinant is well-defined.

Conclusion

Calculating higher-order determinants requires patience — or a good calculator. But understanding the logic behind them, such as Laplace expansion and the permutation-based definition, prepares us to use matrices confidently in any scientific field.

🧩 Supplement: What is a Permutation?

A permutation is a rearrangement of the elements of a set. For example, the permutations of the set {1, 2, 3} are:
(1,2,3), (1,3,2), (2,1,3), (2,3,1), (3,1,2), and (3,2,1). There are 6 permutations in total because 3! = 6.

Given n elements, one of the permutations is the identity permutation. If two elements are out of order, we say there was an inversion. If the number of inversions is even, the sign is positive; if odd, the sign is negative.

For example, the identity permutation (1, 2, 3) has a positive sign; (1, 3, 2) has a negative sign because 2 and 3 are inverted. (2,1,3) has one inversion — negative; (2,3,1) has two inversions — positive; (3,1,2) — two inversions — positive; (3,2,1) — three inversions — negative.

In the case of determinants, we deal with permutations of column (or row) indices. Each term in the formal definition of the determinant has a sign based on the number of inversions in its indices — the parity.

📘 In the next post: Properties of determinants — how they can speed up calculations and help avoid pitfalls.