System of linear equations with parameters, using a matrix Let there be the following system of lin

Tristian Velazquez

Tristian Velazquez

Answered question

2022-06-13

System of linear equations with parameters, using a matrix
Let there be the following system of linear equations:
x + z + b w = a a x + y + a z + ( a + a b ) w = 1 + a 2 b x + ( a + b ) z + ( 1 + b 2 ) w = 4 + a b x + b z + ( a a b + b 2 ) w = a + 1 + a b
a,b parameters. The question is, for which a,b there is no solution to the system, for which there are infinite and for which there is one. I put it into a matrix and with some row operations I got to:
1 0 1 b a 0 1 0 a 1 0 0 a 1 4 0 0 0 a 2 b a + 1
How do I continue from here? I'm quite confused. For instance I thought that saying that for all b, if a=2b then there is no solution.

Answer & Explanation

last99erib

last99erib

Beginner2022-06-14Added 19 answers

You've done good until that row reduced form
Now, if a = 0 you can do another row operation:
[ 1 0 1 b 0 0 1 0 0 1 0 0 0 1 4 0 0 0 2 b 1 ] [ 1 0 1 b 0 0 1 0 0 1 0 0 0 1 4 0 0 0 0 1 + 8 b ]
What can you say about this case?
If, instead, a 0, you have to distinguish when a 2 b = 0 or not.
If a 2 b = 0 you have a = 2 b and you have a solution if and only if a + 1 = 0, that is a = 1 and b = 2.
If a 2 b 0, then...
Feinsn

Feinsn

Beginner2022-06-15Added 8 answers

Hint: Calculate the determinant of the matrix of coefficients and find those a,b for which it equals 0.

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