How to determine the rank of a linear system of equations involving parameters?For instance, the following matrix: ( k 1 1 1 k 1 1 1 k )

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How to determine the rank of a linear system of equations involving parameters?For instance, the following matrix:      (                            k                          1                          1                                      1                          k                          1                                      1                          1                          k                      )

necrologo9yh43 · Accepted Answer

For your example problem, I would try looking at the determinant of the matrix for the system of equations. So, in this case, we can calculate the determinant as       k    3    −  3  k  +  2. Factoring, we see this has two distinct roots,   k  =  1 and   k  =  −  2. Now, if   k is not a root, then the determinant is non-zero and thus the system has full rank. We need to examine the   k  =  1  ,  −  2 cases separately then: if   k  =  1, all the columns are the same, and thus the system can only have rank   1. If   k  =  −  2, note that any two of the columns are not related by scalar multiplication, and hence independent. Thus, the system has rank   2.In general, examining the determinant is often a useful way to approach these problems; it gives us a fairly immediate test for being full rank, and by finding where the determinant vanishes we can know where the system is below full rank. At this point, I&#039;d say just plug those values in and try Gaussian elimination.

How to determine the rank of a linear system of equations involving parameters? For instance, the f

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