The matrices listed in Eq. (11) are used in some of the exercises that follow. 13 ^-[(12]· --[123] -- [(122] A = B = = D = F = 100 010 010 121 032 001 -[ 010 002 013 In Exercises 16-27, use Definition 12 to determine whether the given matrix is singular or nonsingular. If a matrix M is singular, give all solutions of Mx = 0. 16. A 17. B 18. C 19. AB 20. BA 21. D 22. F 23. D + F 25. EF 26. DE 24. E 27. FT

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Chapter2: Second-order Linear Odes
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Linear algebra: please solve q22 and 25 correctly and handwritten. Definition is also attached
The matrices listed in Eq. (11) are used in some of the
exercises that follow.
2
^-[13]· *-[23]· <-[22].
A =
B =
=
34
-[-]
E =
D=
100
010
010
1
-
F = 0 32
001
010
16. A
19. AB
22. F
25. EF
002
013
In Exercises 16-27, use Definition 12 to determine
whether the given matrix is singular or nonsingular. If a
matrix M is singular, give all solutions of Mx = 0.
17. B
18. C
20. BA
21. D
23. D + F
26. DE
24. E
27. FT
Transcribed Image Text:The matrices listed in Eq. (11) are used in some of the exercises that follow. 2 ^-[13]· *-[23]· <-[22]. A = B = = 34 -[-] E = D= 100 010 010 1 - F = 0 32 001 010 16. A 19. AB 22. F 25. EF 002 013 In Exercises 16-27, use Definition 12 to determine whether the given matrix is singular or nonsingular. If a matrix M is singular, give all solutions of Mx = 0. 17. B 18. C 20. BA 21. D 23. D + F 26. DE 24. E 27. FT
DEFINITION 12
An (n × n) matrix A is nonsingular if the only solution to Ax = 0 is x = 0.
Furthermore, A is said to be singular if A is not nonsingular.
If A = [A₁, A2, ..., A₂], then Ax = 0 can be written as
X₁A₁ + x₂A₂ + + x₂A₂ = 0,
so it is an immediate consequence of Definition 12 that A is nonsingular if and only if
the column vectors of A form a linearly independent set. This observation is important
enough to be stated as a theorem.
Transcribed Image Text:DEFINITION 12 An (n × n) matrix A is nonsingular if the only solution to Ax = 0 is x = 0. Furthermore, A is said to be singular if A is not nonsingular. If A = [A₁, A2, ..., A₂], then Ax = 0 can be written as X₁A₁ + x₂A₂ + + x₂A₂ = 0, so it is an immediate consequence of Definition 12 that A is nonsingular if and only if the column vectors of A form a linearly independent set. This observation is important enough to be stated as a theorem.
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