Question

Find a square matrix A of order 2 × 2 such that A² = I₂. 

Answer 1

 Let A = [(a, b), (c, d)] be the required matrix.

Then, A² = I. So

Comparing respective entries we get

a² + bc = 1 (1)

b + bd = 0 (2)

ac + cd = 0 (3)

cb + d² = 1 (4)

These must hold simultaneously.

If α + d = 0, the above four equations hold simultaneously if d = -a and a² + bc = 1.

Hence, one possible square root of I is

 where α, β, γ are the three numbers related by the condition α² + βγ = 1.

If a + d ≠ 0, then the above four equations hold simultaneously if b = 0, c = 0, a = 1, d = 1 or if b = 0, c = 0, a = -1, d = -1.

Hence

i.e. ±I are the values of A.