Determine the value of a if the system x 1 + 4 x 2 −...

dikcijom2k

dikcijom2k

Answered

2022-07-08

Determine the value of a if the system
x 1 + 4 x 2 3 x 3 + 2 x 4 = 0 2 x 1 + 7 x 2 4 x 3 + 4 x 4 = 0 x 1 + a x 2 + 5 x 3 2 x 4 = 0 3 x 1 + 10 x 2 5 x 3 + ( a ² + 4 a + 1 ) x 4 = 0
has more then 1 solution

Answer & Explanation

Tanner Hamilton

Tanner Hamilton

Expert

2022-07-09Added 12 answers

We know that a homogeneous linear system has more than one solution if and only if the rank of the matrix of the system is less than the number of unknowns. See Rank-nullity theorem.
One of the possibilities how to find rank is to use row operations. You could start like this:
( 1 4 3 2 2 7 4 4 1 a 5 2 3 10 5 a 2 + 4 a + 1 ) ( 1 4 3 2 0 1 2 0 0 a + 4 2 0 3 10 5 a 2 + 4 a + 1 ) ( 1 4 3 2 0 1 2 0 0 a + 4 2 0 3 10 5 a 2 + 4 a + 1 ) ( 1 4 3 2 0 1 2 0 0 a + 5 0 0 3 10 5 a 2 + 4 a + 1 )
If a + 5 = 0, i.e. if a = 5, then you have a matrix which does not have full rank. So in this case you have that there is more than one solution.
If a + 5 0, you can divide the third row by ( a + 5 )
( 1 4 3 2 0 1 2 0 0 a + 5 0 0 3 10 5 a 2 + 4 a + 1 ) ( 1 4 3 2 0 1 2 0 0 1 0 0 3 10 5 a 2 + 4 a + 1 ) ( 1 0 3 2 0 0 2 0 0 1 0 0 3 0 5 a 2 + 4 a + 1 ) ( 1 0 3 2 0 1 0 0 0 0 1 0 3 0 5 a 2 + 4 a + 1 ) ( 1 0 0 2 0 1 0 0 0 0 1 0 3 0 0 a 2 + 4 a + 1 )

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