Question

How many 2×2 matrices A satisfyboth A3 = I2 and A2 = At,where I2 denotesthe2×2identity matrix and At denotes the transpose of A?

Answer 1

A3=IA3=I implies A2A=AA2=IA2A=AA2=I using associativity of matrix multiplication.
Using A2=ATA2=AT, we get ATA=AAT=IATA=AAT=I.
So AA must be of the form [cosθsinθ−sinθcosθ][cos⁡θ−sin⁡θsin⁡θcos⁡θ] or [cosθsinθsinθ−cosθ][cos⁡θsin⁡θsin⁡θ−cos⁡θ] (See here for proof.)

Now substitute this into the two given equations to get equations in terms of θθ and solve for θθ.