Linear Algebra Exercises-n-Answers.pdf

Answers to Exercises 27
One.III.1.10
Routine Gauss’ method gives one:
⎛
−3ρ 1+ρ 2
−→ ⎝ 2 1 1 3
⎞
0 1 −2 −7
−(1/2)ρ 1 +ρ 3
0 9/2 1/2 7/2
⎠ −(9/2)ρ2+ρ3
−→
and any cosmetic change, like multiplying the bottom row by 2,
⎛
⎝ 2 1 1 3 ⎞
0 1 −2 −7⎠
0 0 19 70
gives another.
⎛
⎝ 2 1 1 3
⎞
0 1 −2 −7⎠
0 0 19/2 35
One.III.1.11 In the cases listed below, we take a, b ∈ R. Thus, some canonical forms listed below
actually ( include ) ( infinitely ) ( many)
cases. ( In)
particular, they includes the cases a = 0 and b = 0.
0 0 1 a 0 1 1 0
(a) , , ,
0 0 0 0 0 0 0 1
( ) ( ) ( ) ( ) ( ) ( ) ( )
0 0 0 1 a b 0 1 a 0 0 1 1 0 a 1 a 0 0 1 0
(b)
,
,
,
,
,
,
⎛
0 0 0 0 0 0 0 0 0 0 0 0 0 1 b 0 0 1 0 0 1
(c) ⎝ 0 0
⎞ ⎛
0 0⎠,
⎝ 1 a
⎞ ⎛
0 0⎠,
⎝ 0 1
⎞ ⎛
0 0⎠,
⎝ 1 0
⎞
0 1⎠
0 0 0 0 0 0 0 0
⎛
(d) ⎝ 0 0 0
⎞ ⎛
0 0 0⎠,
⎝ 1 a b
⎞ ⎛
0 0 0⎠,
⎝ 0 1 a
⎞ ⎛
0 0 0⎠,
⎝ 0 0 1
⎞ ⎛
0 0 0⎠,
⎝ 1 0 a
⎞ ⎛
0 1 b⎠,
⎝ 1 a 0
⎞ ⎛
0 0 1⎠,
⎝ 1 0 0
⎞
0 1 0⎠
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
One.III.1.12 A nonsingular homogeneous linear system has a unique solution. So a nonsingular matrix
must reduce to a (square) matrix that is all 0’s except for 1’s down the upper-left to lower-right diagonal,
e.g.,
⎛
( )
1 0
, or ⎝ 1 0 0 ⎞
0 1 0⎠ , etc.
0 1
0 0 1
One.III.1.13 (a) The ρ i ↔ ρ i operation does not change A.
(b) For instance, ( ) ( ) ( )
1 2 −ρ 1+ρ 1 0 0 ρ 1+ρ 1 0 0
−→ −→
3 4 3 4 3 4
leaves the matrix changed.
(c) If i ≠ j then
⎛
⎞
.
.
a i,1 · · · a i,n
.
⎜
⎝a j,1 · · · a j,n
⎟
⎠
.
kρ i +ρ j
−→
−kρ i +ρ j
−→
⎛
⎞
.
.
a i,1 · · · a i,n
.
⎜
⎝ka i,1 + a j,1 · · · ka i,n + a j,n
⎟
⎠
.
⎛
.
.
a i,1 · · · a i,n
.
⎜
⎝−ka i,1 + ka i,1 + a j,1 · · · −ka i,n + ka i,n + a j,n
⎟
⎠
.
does indeed give A back. (Of course, if i = j then the third matrix would have entries of the form
−k(ka i,j + a i,j ) + ka i,j + a i,j .)
⎞
Subsection One.III.2: Row Equivalence
One.III.2.11
Bring each to reduced echelon form and compare.