Determinants were born to solve systems of linear equations. In 1750 Gabriel Cramer published the rule that bears his name, which gives each unknown as a ratio of two expressions formed from the coefficients. But an organized way of computing those expressions when there were many unknowns was missing. In 1772, almost at the same time, Alexandre-Théophile Vandermonde and Laplace studied these expressions as objects in their own right. The word determinant, in its present sense, was fixed only by Cauchy in 1812, and the vertical bars we use today came from Arthur Cayley in 1841.

Six terms, three groups

The determinant of a $3 \times 3$ matrix has six terms:

$$\begin{vmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{vmatrix} = a_1b_2c_3 - a_1b_3c_2 - a_2b_1c_3 + a_2b_3c_1 + a_3b_1c_2 - a_3b_2c_1.$$

Laplace’s idea is to group these terms by the entries of a row (or column). Factoring out $a_1$, $a_2$ and $a_3$, the entries of the first column:

$$= a_1(b_2c_3 - b_3c_2) - a_2(b_1c_3 - b_3c_1) + a_3(b_1c_2 - b_2c_1).$$

Each bracket is a $2 \times 2$ determinant: what is left when the row and column of the entry are deleted. This smaller determinant is called the minor of the entry:

$$= a_1\begin{vmatrix} b_2 & c_2 \\ b_3 & c_3 \end{vmatrix} - a_2\begin{vmatrix} b_1 & c_1 \\ b_3 & c_3 \end{vmatrix} + a_3\begin{vmatrix} b_1 & c_1 \\ b_2 & c_2 \end{vmatrix}.$$

The chessboard of signs

The signs alternate in a chessboard pattern: the entry in row $i$ and column $j$ carries the sign $(-1)^{i+j}$.

$$\begin{pmatrix} + & - & + \\ - & + & - \\ + & - & + \end{pmatrix}$$

The minor multiplied by this sign is called the cofactor. Laplace’s expansion says that the determinant of a matrix of any size is the sum of the entries of any row or column, each multiplied by its cofactor. The theorem Laplace proved is even more general: instead of one row, one can choose several rows at once and use larger minors.

Choosing the row well

Since any row or column may be chosen, it pays to choose the one with the most zeros. For example:

$$\begin{vmatrix} 2 & 1 & 3 \\ 0 & 4 & 1 \\ 0 & 2 & 5 \end{vmatrix} = 2\begin{vmatrix} 4 & 1 \\ 2 & 5 \end{vmatrix} - 0 + 0 = 2(20 - 2) = 36.$$

Expanding along the first column, only one term survives.

Beautiful, but slow

For an $n \times n$ matrix, the full expansion produces $n!$ terms. For $n = 10$ there are $3\,628\,800$; for $n = 20$, about $2.4 \times 10^{18}$. That is why, for large matrices, nobody computes determinants this way: Gaussian elimination does the job in about $\frac{n^3}{3}$ operations. Laplace’s expansion remains precious for small matrices, for matrices full of zeros and, above all, for proving properties of determinants.

Exercise: a 4×4 determinant

Compute

$$\begin{vmatrix} 1 & 0 & 2 & 0 \\ 3 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 2 & 5 & 4 & 2 \end{vmatrix}.$$

The last column has only one nonzero entry, the 2 in position (4, 4), with sign $(-1)^{8} = +$. Expanding along it:

$$2\begin{vmatrix} 1 & 0 & 2 \\ 3 & 1 & 0 \\ 0 & 0 & 1 \end{vmatrix}.$$

Now the last row has only the 1 in position (3, 3), with sign $+$: $2 \cdot 1 \cdot \begin{vmatrix} 1 & 0 \\ 3 & 1 \end{vmatrix} = 2 \cdot (1 - 0) = 2$.

Exercise: two equal rows

Use Laplace’s expansion to show that a $3 \times 3$ determinant with two equal rows is zero.

Suppose rows 1 and 2 are equal and expand along row 3, the remaining one. Each cofactor is, up to sign, a $2 \times 2$ determinant formed from rows 1 and 2, which are equal: $\begin{vmatrix} p & q \\ p & q \end{vmatrix} = pq - qp = 0$. All the terms vanish and the determinant is zero. The same argument, by induction, works for any size.

Sources: Pierre-Simon Laplace, “Recherches sur le calcul intégral et sur le système du monde”, Mémoires de l’Académie royale des sciences (1772, published 1776) · Alexandre-Théophile Vandermonde, “Mémoire sur l’élimination”, Mémoires de l’Académie royale des sciences (1772) · Augustin-Louis Cauchy, “Mémoire sur les fonctions qui ne peuvent obtenir que deux valeurs égales et de signes contraires”, Journal de l’École polytechnique 10 (1815) · Thomas Muir, The Theory of Determinants in the Historical Order of Development, vol. 1 (1906).

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