Doubt from matrices

In illustration 27 how did they got A^2=4A-5I, i mean without giving this how can i get to know the relation between A^2 in terms of A and I ?

You have to analyze
You always can write A^2=aA+bI

For a 2X2 matrix it is useful to remember that

A^2- tr(A) A+|A| I=0 where tr(A) is the sum of the diagonal elements and |A| is determinant.

Hence here we get A^2-4A+5I=0

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Thank you sir