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 − 3I 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.

Question

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  −  3I 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&#039;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&#039;m missing something. Any help is appreciated.

timmeraared · Accepted Answer

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.

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

Answered question

Answer & Explanation