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Originally posted in a technical math section, so missing the template

Let S = [ x x x x ; a x x x ; 0 b x x ; 0 0 c x]

Find the rank of S-K*identity.

Attempt I basically did straight forward row reduction and got

[ (x-k) x x x ; 0 ((x-k)^2-ax)/(x-k) x-ax/(1-k) a- ax/(1-k); 0 0 (x-k)-b[x(x-k)-ax]/ [(x-k)^2-ax] (x-k)-b[x(x-k)-ax]/[(x-k)^2-ax]; 0 0 0 [(x-k)^2-ax(x-k)-c]/[(x-k)^2 -ax] ]

It is given that the rank is 3. I am not sure how to prove this. Shouldn't I have gotten a zero row when row reducing?

Are there any alternatives to find the rank of a matrix like finding the upper bound of the rank is 3 and the lower bound for the rank is 3?

Find the rank of S-K*identity.

Attempt I basically did straight forward row reduction and got

[ (x-k) x x x ; 0 ((x-k)^2-ax)/(x-k) x-ax/(1-k) a- ax/(1-k); 0 0 (x-k)-b[x(x-k)-ax]/ [(x-k)^2-ax] (x-k)-b[x(x-k)-ax]/[(x-k)^2-ax]; 0 0 0 [(x-k)^2-ax(x-k)-c]/[(x-k)^2 -ax] ]

It is given that the rank is 3. I am not sure how to prove this. Shouldn't I have gotten a zero row when row reducing?

Are there any alternatives to find the rank of a matrix like finding the upper bound of the rank is 3 and the lower bound for the rank is 3?

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