Math 308 Section-Zhou,
Review for Exam 3
Linear Algebra ((a)-(c), Review)
(a) To solve Ax = b by augmented matrix [A|b] using three elementary row operations (3ERO):
(1) (i) ↔ (j), (2) α(i), (3) α(i)+
→ (j) (Elimination method). To be able to determine if Ax = b has
(1) no solution, (2) unique solution or (3) infinite solutions. In cases of (2)/(3), find all solutions.

(b) Find A−1 , A−1
2×2

a22 −a12 
= d1 
, (d = |A|) or [A|I] → · · · → [I|A−1 ] using 3ERO.
−a21 a11
(c) Evaluate |A|. n = 2, 3, use the formula. n = 4, use cofactor expansion along ith-row/jth-column:
|A| =
n
∑
(−1)i+k aik |Mik | =
k=1
n
∑
(−1)k+j akj |Mkj |, use properties or 3ERO (−1, α, 1) or combination.
k=1
(d) Find all E-values/E-vectors: Av = λv, (v ̸= 0). (1) Solve |A − λI| = 0 for λ1 , ..., λn . (2) For λi ,
solve (A − λi I)vi = 0, ([A − λi I|0]) for vi . (3) treat parameter(s). Real, complex, non-defective and
defective.
(e) Solve linear system X ′ (t) = AX(t). Solve Avi = λi vi .
(1) Real, distinct or repeated but non-defective, Xi (t) = eλi t vi , X(t) = c1 X1 (t) + · · · + cn Xn (t),
(2) Complex E-value λ = µ + iν, v = ⃗a + i⃗b, two real-valued solutions:
X1 (t) = eµt [cos(νt)⃗a − sin(νt)⃗b], X2 (t) = eµt [cos(νt)⃗b + sin(νt)⃗a].
(3) Real repeated and defective: X1 (t) = eλt v, X2 (t) = eλt (tv + u) where (A − λI)u = v.
(4) Using I.C. X(0) = X0 to determine c1 , ..., cn by solving Ψ(0)C = X0 or [Ψ(0)|X0 ] for C
where Ψ(0) = [X1 (0) · · · Xn (0)], the fundamental matrix at t = 0. (Did not cover)
(f) Convert a high order equation to a 1st-order system. (did not cover)
= AX, |A| ̸= 0. The only critical point Xc = 0. Find eigenvalues r1 , r2 of A.
(g) Linear System dX
dt
(a) r1 ̸= r2 real, (b) r1 = r2 , (c) r = µ ± iν. Determine (in)stability at Xc :
AS if r1 , r2 < 0 in (a), r < 0 in (b), µ < 0 in (c); US if r1 , r2 > 0 in (a), r > 0 in (b), µ > 0 in (c);
US saddle if r1 < 0, r2 > 0 in (a) (determine its directions of motion); Stable center if µ = 0 in (c).

(h) Locally Linear System (LLS)


dX
dt



Fx Fy 
F (x, y) 
= f (X) = 
. Its Jacobian J = f ′ (X) = 
.
G(x, y)
Gx Gy


F (x, y)   0 
(1) Solve f (X) = 
=
for all critical point(s) Xc .
G(x, y)
0


Fx Fy 
(2) For each Xc , evaluate its Jacobian matrix A = f ′ (Xc ) = 
Gx Gy
X=Xc
.
(3) Determine (in)stability of its corresponding linear system (CLS) dU
= AU at U = 0 from (g).
dt
(4) LLS at Xc and CLS at U = 0 have the same (in)stability except the case r = ±iν where CLS
has a stable center at Xc but LLS is inconclusive.
(5) Competing-species-model (US,US saddle,AS) and Predator-prey-model (US saddle, Stable cyclic).
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