Consider the matrix A = [[2, 0, 0],[1,1,-2],[1,-1, 0]]   (b) For which values of k is A-kI not invertible? These are the Eigenvalues of A.

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A = [[2, 0, 0],[1,1,-2],[1,-1, 0]]

`A-kI=[[2-k,0,0],[1,1-k,-2],[1,-1,-k]]`

`A-kI `   is not invertible if  det(A-kI)=0 ,

Yes , k is the eigen value of matrix A.

`|A-kI|=(2-k)(-k+k^2-2)` =0

either  2-k=0

k=2

or

`k^2-k-2=0`

`k^2-2k+k-1=0`

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

`k=2,-1`

Thus if K=2,-1 then matrix A-kI will not invertible.

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