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Suppose A is a 2×2 matrix such that A2 = I2 and let J be the...

Suppose A is a 2×2 matrix such that A2 = I2 and let J be the Jordan form of A. Show that J2 = I2 and use this fact to conclude that J is diagonal

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Answer #1

we have given that A2=I where A is a 22 matrix and also I is the identity matrix of order 2

as we know A2=I imply A is involutary matrix

and a involutary matrix has eigen value 1 or -1 as its determinant is equal to 1 or -1 by above condition

A2=I imply (A-I)(A+I)=0 imply either |A-I|=0 OR |A+I|=0

hence its eigen values are 1 and -1 and its jordan form will be

J=    where J represent the jordan form of A

and J2==  

HENCE we get J2=I where J is the diagonal matrix and I is the identity matrix of order 2

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