Effect of multiplying by regular matrix on number of solutions of linear equations
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I have following problem:
Let $A,B in F^{ntimes n}$, $B$ is regular. Which on of the following is true?
1.) If $(A|b)$ has at least one solution, then $(BA|b)$ has at least one solution.
2.) If $(A|b)$ has at least one solution, then $(AB|b)$ has at least one solution.
3.) If $(BA|b)$ has at least one solution, then $(A|b)$ has at least one solution.
4.) If $(AB|b)$ has at least one solution, then $(A|b)$ has at least one solution.
($(A|b)$ is a set of linear equations, not sure if that's clear)
If my logic is correct, 1 and 3 are true, and 2 and 4 are false.
Because regular matrix cannot change number of solutions, the multiplication should have no effect in this implication. However this is not true the other way around, if we multiply with not regular matrix, there is a possibility, that the number of solutions may change.
However the answer feels way too easy, so i am not sure if i am missing something obvious here.
Thanks for your time!
linear-algebra matrices matrix-equations
asked Nov 14 at 5:50
Tomáš Korený
32
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up vote
0
down vote
favorite
I have following problem:
Let $A,B in F^{ntimes n}$, $B$ is regular. Which on of the following is true?
1.) If $(A|b)$ has at least one solution, then $(BA|b)$ has at least one solution.
2.) If $(A|b)$ has at least one solution, then $(AB|b)$ has at least one solution.
3.) If $(BA|b)$ has at least one solution, then $(A|b)$ has at least one solution.
4.) If $(AB|b)$ has at least one solution, then $(A|b)$ has at least one solution.
($(A|b)$ is a set of linear equations, not sure if that's clear)
If my logic is correct, 1 and 3 are true, and 2 and 4 are false.
Because regular matrix cannot change number of solutions, the multiplication should have no effect in this implication. However this is not true the other way around, if we multiply with not regular matrix, there is a possibility, that the number of solutions may change.
However the answer feels way too easy, so i am not sure if i am missing something obvious here.
Thanks for your time!
linear-algebra matrices matrix-equations
asked Nov 14 at 5:50
Tomáš Korený
32
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I have following problem:
Let $A,B in F^{ntimes n}$, $B$ is regular. Which on of the following is true?
1.) If $(A|b)$ has at least one solution, then $(BA|b)$ has at least one solution.
2.) If $(A|b)$ has at least one solution, then $(AB|b)$ has at least one solution.
3.) If $(BA|b)$ has at least one solution, then $(A|b)$ has at least one solution.
4.) If $(AB|b)$ has at least one solution, then $(A|b)$ has at least one solution.
($(A|b)$ is a set of linear equations, not sure if that's clear)
If my logic is correct, 1 and 3 are true, and 2 and 4 are false.
Because regular matrix cannot change number of solutions, the multiplication should have no effect in this implication. However this is not true the other way around, if we multiply with not regular matrix, there is a possibility, that the number of solutions may change.
However the answer feels way too easy, so i am not sure if i am missing something obvious here.
Thanks for your time!
linear-algebra matrices matrix-equations
asked Nov 14 at 5:50
Tomáš Korený
32
I have following problem:
Let $A,B in F^{ntimes n}$, $B$ is regular. Which on of the following is true?
1.) If $(A|b)$ has at least one solution, then $(BA|b)$ has at least one solution.
2.) If $(A|b)$ has at least one solution, then $(AB|b)$ has at least one solution.
3.) If $(BA|b)$ has at least one solution, then $(A|b)$ has at least one solution.
4.) If $(AB|b)$ has at least one solution, then $(A|b)$ has at least one solution.
($(A|b)$ is a set of linear equations, not sure if that's clear)
If my logic is correct, 1 and 3 are true, and 2 and 4 are false.
Because regular matrix cannot change number of solutions, the multiplication should have no effect in this implication. However this is not true the other way around, if we multiply with not regular matrix, there is a possibility, that the number of solutions may change.
However the answer feels way too easy, so i am not sure if i am missing something obvious here.
Thanks for your time!
linear-algebra matrices matrix-equations
linear-algebra matrices matrix-equations
asked Nov 14 at 5:50
Tomáš Korený
32
asked Nov 14 at 5:50
Tomáš Korený
32
asked Nov 14 at 5:50
Tomáš Korený
32
asked Nov 14 at 5:50
Tomáš Korený
32
asked Nov 14 at 5:50
Tomáš Korený
32
32
add a comment |
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