Use the result of Problem 1 to find all values of c for which the homogeneous system
(c + 2) x + 3y = 0
2x + (c − 3) y = 0
has a nontrivial solution.
Problem 1
Show that the homogeneous system in Problem 2 has a nontrivial solution if and only if ad – bc = 0.
Problem 2
Consider the homogeneous system
ax + by = 0
cx + dy = 0.
(a) If x = x0 and y = y0 is a solution and k is a real number, then show that x = kx0 and y = ky0 is also a solution.
(b) If x = x1, y = y1 and x = x2, y = y2 are both solutions, then show that x = x1 + x2, y = y1 + y2 is a solution.
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