(1) Derive, as cleanly as possible, the determinant formula for the
solution of the system


a11 x1 + a12 x2 + a13 x3 = b1
a21 x1 + a22 x2 + a23 x3 = b2

a x + a x + a x = b
31 1
32 2
33 3
3
Hint: It should be
x1 =
det(A1 )
,
det(A)
where


a11 a12 a13
A = a21 a22 a23 
a31 a32 a33


b1 a12 a13
A1 = b2 a22 a23  .
b3 a32 a33
(2) In class, we derived two forms for the general solution of
y 00 + 2y + 5y = 0,
namely
y(t) = Ae(−1+2i)t + Be(−1−2i)t
and
y(t) = C1 e−t cos(2t) + C2 e−t sin(2t).
Find the constants {A, B} and {C1 , C2 } corresponding to the
initial conditions y(0) = 0 and y 0 (0) = 1. Verify that they are
related by
C2 = i(A − B).
C1 = A + B,
Observe that A and B are complex conjugates of each other.
1