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Linear Algebra Problem Set 2.3~2.4
2.3 Elimination Using Matrices (27)
27. Choose the numbers a, b, c, d in this augmented matrix so that there is:
(a) No solution
(b) Infinitely many solutions
(c) Which of the numbers a, b, c, or d have no effect on the solvability?
⎡1 2 3 a ⎤
[ A b] = ⎢⎢0 4 5 b ⎥⎥
⎢⎣0 0 d c ⎥⎦
Sol.
(a) d = 0 and c ≠ 0
(b) c = 0 and d = 0
(c) a and b
Recitation 2 ‐ 2008/09/30 1
Linear Algebra Problem Set 2.3~2.4
2.4 Lengths and Dot Products (2.4 C, 10, 26, 27, 33, 34)
2.4 C A directed graph starts with n nodes. There are n2 possible edges-each edge
leaves one of the n nodes and enters one of the n nodes (possibly itself). The n by n
adjacency matrix has aij = 1 when an edge leaves node i and enters node j; if no edge
then aij = 0. Here are two directed graphs and their adjacency matrices:
⎡1 1 ⎤
A=⎢
⎥
⎣1 0⎦
⎡1 1 1 ⎤
⎢1 0 0⎥
⎢
⎥
⎢⎣1 0 0⎥⎦
(a) List all of the 3-step paths between each pair of nodes and compare with A3.
(b) What is the diameter of the second graph?
Sol.
⎡3 2⎤
A3 = ⎢
⎥
⎣2 1 ⎦
(a)
⎡1 to 1 to 1 to 1, 1 to 2 to 1 to 1, 1 to 1 to 2 to 1
⎢
2 to 1 to 2 to 1, 2 to 1 to 1 to 1
⎣
1 to 1 to 1 to 2, 1 to 2 to 1 to 2⎤
⎥
2 to 1 to 1 to 2
⎦
⎡1 1 1 ⎤ ⎡1 1 1 ⎤ ⎡3 1 1⎤
(b) A = ⎢⎢1 0 0 ⎥⎥ ⎢⎢1 0 0 ⎥⎥ = ⎢⎢1 1 1⎥⎥
, the second graph has diameter 2.
⎢⎣1 0 0⎥⎦ ⎢⎣1 0 0 ⎥⎦ ⎢⎣1 1 1⎥⎦
2
10. Row 1 of A is again added to row 2 to produce EA. Then F adds row 2 of EA to
row 1. The result is F(EA):
b ⎤ ⎡ 2a + c 2b + d ⎤
⎡a b ⎤
⎡1 1⎤ ⎡ a
A=⎢
, F ( EA) = ⎢
⎥
⎥
⎢
⎥=⎢
⎥
⎣c d ⎦
⎣0 1⎦ ⎣ a + c b + d ⎦ ⎣ a + c b + d ⎦
(a) Do those steps in the opposite order: first add row 2 to row 1 by FA, then add
row 1 of FA to row 2.
(b) What law is or is not obeyed by matrix multiplication?
Sol.
⎡1 0⎤ ⎡ a + c b + d ⎤ ⎡ a + c b + d ⎤
=
(a) E ( FA) = ⎢
⎥⎢
d ⎥⎦ ⎢⎣ a + 2c b + 2d ⎥⎦
⎣1 1 ⎦ ⎣ c
(b) Commutative law
Recitation 2 ‐ 2008/09/30 2
Linear Algebra Problem Set 2.3~2.4
26. Multiply AB using columns times rows:
⎡1 0 ⎤
⎡1 ⎤
⎡3 3 0 ⎤ ⎢ ⎥
⎢
⎥
AB = ⎢ 2 4⎥ ⎢
= 2 [ 3 3 0] +
1 2 1 ⎥⎦ ⎢ ⎥
⎣
⎢⎣ 2 1 ⎥⎦
⎢⎣ 2 ⎥⎦
=
Sol.
⎡1
AB = ⎢⎢ 2
⎢⎣ 2
⎡3
= ⎢⎢6
⎢⎣6
0⎤
⎡1 ⎤
⎡0 ⎤
⎡3 3 0 ⎤ ⎢ ⎥
⎥
= ⎢ 2⎥ [3 3 0] + ⎢⎢ 4 ⎥⎥ [1 2 1]
4⎥ ⎢
⎥
1 2 1⎦
⎢⎣ 2⎥⎦
⎢⎣1 ⎥⎦
1 ⎥⎦ ⎣
3 0⎤ ⎡0 0 0 ⎤ ⎡ 3 3 0 ⎤
6 0⎥⎥ + ⎢⎢ 4 8 4⎥⎥ = ⎢⎢10 14 4⎥⎥
6 0⎥⎦ ⎢⎣1 2 1 ⎥⎦ ⎢⎣ 7 8 1 ⎥⎦
27. The product of upper triangular matrices is always upper triangular:
⎡ x x x⎤ ⎡ x x x⎤ ⎡
AB = ⎢⎢ 0 x x ⎥⎥ ⎢⎢ 0 x x ⎥⎥ = ⎢⎢0
⎢⎣ 0 0 x ⎥⎦ ⎢⎣ 0 0 x ⎥⎦ ⎢⎣0 0
⎤
⎥
⎥
⎥⎦
(a) Row times column is dot product (Row 2 of A)⋅ (Column 1 of B) = 0. Which
other dot products give zeros?
(b) Column times row is full matrix Draw x’s and 0’s in (column 2 of A) times
(row 2 of B) and in (column 3 of A) times (row 3 of B).
Sol.
(a) (Row 3 of A)⋅ (Column 1 of B) and (Row 3 of A)⋅ (Column 2 of B)
⎡ x⎤
⎡0 x x ⎤
⎡ x⎤
⎡0 0 x ⎤
⎢
⎥
⎢
⎥
⎢
⎥
x [ 0 0 x ] = ⎢⎢0 0 x ⎥⎥
(b) ⎢ x ⎥ [ 0 x x ] = ⎢0 x x ⎥
, ⎢ ⎥
⎢⎣ 0 ⎥⎦
⎢⎣0 0 0 ⎥⎦
⎢⎣ x ⎥⎦
⎢⎣0 0 x ⎥⎦
Recitation 2 ‐ 2008/09/30 3
Linear Algebra Problem Set 2.3~2.4
33. Suppose you solve Ax=b for three special right sides b:
⎡1 ⎤
⎡0 ⎤
⎡0 ⎤
⎢
⎥
⎢
⎥
Ax1 = ⎢0 ⎥ , Ax2 = ⎢1 ⎥ , Ax3 = ⎢⎢0⎥⎥
⎢⎣0 ⎥⎦
⎢⎣0⎥⎦
⎢⎣1 ⎥⎦
If the three solutions x1, x2, x3 are the columns of a matrix X, which is A times X?
Sol.
AX = [ Ax1
Ax2
Ax3 ]
34. If the three solutions in Question 33 are x1=(1, 1, 1) and x2=(0, 1, 1) and x3=(0, 0,
1), solve Ax=b when b=(3, 5, 8). Challenge problem: What is A?
Sol.
⎡1⎤ ⎡1 ⎤ ⎡0⎤ ⎡0⎤ ⎡0⎤ ⎡0 ⎤
A ⎢⎢1⎥⎥ = ⎢⎢0⎥⎥ , A ⎢⎢1 ⎥⎥ = ⎢⎢1 ⎥⎥ , A ⎢⎢0⎥⎥ = ⎢⎢0 ⎥⎥
⎢⎣1⎥⎦ ⎢⎣0⎥⎦ ⎢⎣1 ⎥⎦ ⎢⎣0⎥⎦ ⎢⎣1 ⎥⎦ ⎢⎣1 ⎥⎦
⎡1 0 0 ⎤ ⎡1 0 0⎤
⎡ 1 0 0⎤
⎢
⎥
⎢
⎥
A ⎢1 1 0 ⎥ = ⎢0 1 0⎥ , A = ⎢⎢ −1 1 0⎥⎥
⎢⎣1 1 1 ⎥⎦ ⎢⎣0 0 1 ⎥⎦
⎢⎣ 0 −1 1 ⎥⎦
Recitation 2 ‐ 2008/09/30 4