I have a system of linear equations: x − y + 2 z − t...



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I have a system of linear equations:
x y + 2 z t = 1
2 x 3 y z + t = 1
x + ( α 4 ) z = α 3
I have already found that this system has a solution for any value of α. Now I need to find the α for which the matrix of the system has a rank=2. The row echelon form of the matrix looks like this:
[ 1 1 2 1 1 0 1 5 3 3 0 0 α 11 4 α 7 ]
I don't think that the rank of this matrix could be 2 for any value of α but the problem specifically asks for me to prove that it can. Maybe I'm missing something. Any help is appreciated.

Answer & Explanation



Beginner2022-06-20Added 22 answers

It looks like your transformation to row echolon form is correct.
You are correct, there is no choice for α such that the matrix has rank 2 (both the matrix of the coefficients as well as the augmented including the r.h.s.) since there is no choice for α which will make the last row linear dependent of the first two rows, as can be seen in the echelon form.

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