Page 55 - Textos de Matemática Vol. 39
P. 55
TRIANGLE SIZE AND RANK FOR ZERO-NONZERO PATTERNS 45 Note that, mr(P ) = 4 if and only if Q = 0. Consider the equations q11 = q32 =
q21 = q12 = q23 = 0, therefore a35 = −1
a53
a13 = −a14a46
a62 = a13a35 = a14a46 a53
a53 = a24a46, so a62 = a14 a24
a14 = −a24a62 = −a14, which is a contradiction.
We conclude that Q ̸= 0 and mr(P) > 4. By [2, Corollary 3.3] we obtain
mr(P) = 5 and t(7,7,5) ≤ 4.
The next example is also verified in [2, Section 6]. It may also be verified usingLemma1.6,MT(P)=7andP[{1,2,3,4,6,7,9},{2,3,4,5,6,7,8}]=T is a 7-triangle. Since P [{5, 8}, {1, 9}] is not allowed, mr(P ) > 7, and mr(P ) = 8 is the only possibility by Lemma 1.5.
Example 1.10. For the 9-by-9 pattern
⎡⎢ ∗ ∗ 0 0 0 0 0 0 0 ⎤⎥ ⎢ ∗ ∗ 0 0 0 0 ∗ 0 0 ⎥ ⎢ 0 ∗ 0 0 0 0 ∗ ∗ 0 ⎥ ⎢ 0 0 ∗ ∗ 0 0 0 0 0 ⎥
P=⎢0 0 ∗ ∗ 0 0 0 ∗ 0⎥ ⎢ 0 0 0 ∗ 0 0 0 ∗ ∗ ⎥ ⎢0000∗∗000⎥ ⎣0000∗∗00∗⎦
00000∗∗0∗ mr(P) = 8, while MT(P) = 7. Therefore, t(9,9,8) ≤ 7.
We further illustrate the use of Lemma 1.6 to verify, by example, that t(7,7,6) < 6, which implies, by Lemma 1.1, that t(n,n,n − 1) < n − 1 for n ≥ 7. The analysis is similar to that alluded to for Example 1.10.
Example 1.11. Let P be the following 7-by-7 pattern
⎡⎢ ∗ 0 0 0 0 ∗ 0 ⎤⎥ ⎢ ∗ ∗ 0 0 0 ∗ 0 ⎥ ⎢ ∗ ∗ ∗ 0 0 0 ∗ ⎥
P=⎢0 0 ∗ ∗ 0 0 0⎥. ⎢0∗00∗0∗⎥ ⎣0∗∗∗0∗0⎦
0000∗0∗