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Statement

Here (M) is the equation $B=AA^{\mathrm T}=A^{\mathrm T}A$ of Theorem 2, with $B$ the integral matrix having $N+1$ down the main diagonal and ones in all other positions, and an incidence matrix is as defined there.

Theorem 3 (p. 89, quoted). "If a matrix $A$ with non-negative integral elements and of order $n>1$ satisfies the equation (M), where $N\ge2$, then $A$ is an incidence matrix and defines a finite projective plane geometry with $N+1$ points on a line."

For $N\ge2$, with Theorem 2 this makes the existence of a plane with $N+1$ points on a line equivalent to the existence of a matrix $A$ of some order $n>1$ with nonnegative integral entries satisfying (M). The bound $N\ge2$ is needed: for $N=1$ the $3\times3$ matrix with ones on the diagonal and on one cyclic off-diagonal satisfies (M), and there is no plane (the example is checked here; the paper gives none).

Proof pointer

P. 89. An entry greater than $1$ would, by (M), force every other entry in its row and in its column to vanish, and then $AA^{\mathrm T}$ would have a zero entry, which (M) forbids; so $A$ is a $0$--$1$ matrix. Equation (M) with $N\ge2$ then gives (I1)--(I3), and the incidence matrix defines the plane.

Read depth. Claims checked: the theorem was read clause by clause on the page image of the print, and the proof was followed. Nothing here is independently reviewed.

Dependencies

Theorem 2 for the equation (M) and the definition of an incidence matrix. The paper does not use Theorem 3 in its proof of Theorem 1.

Source. R. H. Bruck and H. J. Ryser, The nonexistence of certain finite projective planes, Canad. J. Math. 1 (1949), 88--93, doi:10.4153/CJM-1949-009-2; the edition read is named on the source card.

Bears on

• Problem 723: the problem asks whether every finite projective plane has prime-power order. Theorem 3 shows that a nonnegative integral solution of (M) with $N\ge2$ is a plane with $N+1$ points on a line, so the problem for order $N$ is a question about such matrix solutions; it excludes no order.