For the matrix 1 0 1 0 0 1 0 1 1 0 1 0 0 1 0 1 Find the rank and eigenvalues I k

Question

For the matrix 1 0 1 0 0 1 0 1 1 0 1 0 0 1 0 1 Find the rank and eigenvalues I know the answer is 2 2 0 0, and I'm pretty sure I know how to get the 0 and o values, not sure about the 2 & 2

Explanation / Answer

|A-?I| = 0 where ? is eigen value ....... [ ] = represents rows... Now |A-?I| = |[1-? 0 1 0][0 1-? 0 1][1 0 1-? 0][ 0 1 0 1-? ]| .......Now R1-->R1+R2+R3+R4 .......|A-?I| = |[2-? 2-? 2-? 2-?][0 1-? 0 1][1 0 1-? 0][ 0 1 0 1-? ]| = (2-?)|[1 1 1 1][0 1-? 0 1][1 0 1-? 0][ 0 1 0 1-? ]| ........Now R2-->R2-R4 ......so we have .....|A-?I| =(2-?) |[1 1 1 1][0 -? 0 ?][1 0 1-? 0][ 0 1 0 1-? ]| ........Now take out ? from R2 .......|A-?I| =(2-?)(?) |[1 1 1 1][0 -1 0 1][1 0 1-? 0][ 0 1 0 1-? ]| ......Now R4-->R4+R2.. and R3-->R3-R1....... |A-?I| =(2-?)(?) |[1 1 1 1][0 -1 0 1][0 -1 -? -1][ 0 0 0 2-? ]| .............now take the det of [1 1 1 1][0 -1 0 1][0 -1 -? -1][ 0 0 0 2-? ] .....in which Column1 elements are 1 0 0 0 .......so |[1 1 1 1][0 -1 0 1][0 -1 -? -1][ 0 0 0 2-? ]| = (1)|[ -1 0 1][ -1 -? -1][ 0 0 2-? ]| = (2-?)(?) ...........so |A-?I| =(2-?)(?)(2-?)(?) ...and |A-?I| = 0 so ....(2-?)(?)(2-?)(?) = 0 ..which gives you ? = 0,0,2,2...........

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