Written by Myoungjean Bae and Anakewit(Tete) Boonkasame. Section 3.3. 32.
Main strategy for this problem is to do row operation several times so that we can
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MATH320 Homework 3 Solutions Written by Myoungjean Bae and Anakewit(Tete) Boonkasame Section 3.3 32. Main strategy for this problem is to · ¸ · do row ¸ operation several times so that we can reduce the matrix a b 1 0 to the 2×2 identity matrix . In the procedure, we need to understand how the condition c d 0 1 ad − bc 6= 0 is applied. First of all, if ad − bc 6= 0, then either a or c is nonzero. Without loss of generality, let us assume that a is nonzero, otherwise we can switch rows. · ¸ · ¸ · ¸ (ad−bc)R1 −bR2 (ad − bc)a 0 b a b aR2 −cR1 a −→ −→ 0 ad − bc 0 ad − bc c d · ¸ R2 · ¸ R1 1 0 1 0 a(ad−bc) ad−bc −→ −→ 0 ad − bc 0 1 In particular, the last two row operations are possible WHEN ad − bc 6= 0. · So we proved that the matrix
¸ · a b 1 is row equivalent to c d 0
¸ 0 provided that ad − bc 6= 0. 1
36.Rewrite the system of homogeneous linear equations as · ¸ a b 0 c d 0 ¸ ¸ · 1 0 a b . This means is row equivalent to 0 1 c d that after several times of row operations, we can write · ¸ a b 0 c d 0 ·
If ad − bc 6= 0 then we have shown that the matrix
as
· 1 0
0 1
¸ 0 . 0
Writing back the equations from the matrix right above, we conclude that x=y=0 is the only solution.¤ 37.If ad − bc = 0, then after several times of row operation, we have · ¸ · ¸ · a b 0 several times of row operations (ad − bc)a 0 0 0 −→ = c d 0 0 ad − bc 0 0 1
0 0
0 0
¸
i.e.,
(
0x + 0y = 0 0x + 0y = 0
This means that x, y can be any real number. Therefore, we conclude that the homogeneous system ( ax + by = 0 (1) cx + dy = 0 has nontrivial(nonzero) solution ONLY IF ad − bc = 0. We note that if ad − bc 6= 0, then we have shown from 36 that the homogeneous system(1) has only (x, y) = (0, 0)(trivial solution) as solution.¤
