# For which values of k is this sytem inconsistent? Consider the following system of equations: {(kx+y+z=-8),(x+ky+z=-1),(x+y+kz=9):} You need to remember that a system is inconsistent if its determinant `Delta` of matrix of the system is equal to zero, such that:

`Delta = [(k,1,1),(1,k,1),(1,1,k)]`

`Delta = k^3 + 1 + 1 - k - k - k => Delta = k^3 - 3k + 2`

Since `Delta...

Start your 48-hour free trial to unlock this answer and thousands more. Enjoy eNotes ad-free and cancel anytime.

You need to remember that a system is inconsistent if its determinant `Delta` of matrix of the system is equal to zero, such that:

`Delta = [(k,1,1),(1,k,1),(1,1,k)]`

`Delta = k^3 + 1 + 1 - k - k - k => Delta = k^3 - 3k + 2`

Since `Delta = 0 => k^3 - 3k + 2 = 0`

`k^3 - 2k - k + 2 = 0`

You need to group the terms such that:

`(k^3 - k) - (2k - 2) = 0`

Factoring out k in the first group yields:

`k(k^2 - 1) - 2(k - 1) = 0 => k(k - 1)(k + 1) - 2(k - 1) = 0`

Factoring out `(k - 1)` yields:

`(k - 1)(k^2 + k - 2k + 2) = 0 => {(k = 1),(k^2 - k + 2 != 0):}`

Hence, evaluating k for the system is inconsistent yields `k = 1` .

Approved by eNotes Editorial Team