Matrices

Let us compute A²:
A = [ 2 -2 ]
[ -2 2 ]
⇒ A² = A ⋅ A = [ 2 -2 ] ⋅ [ 2 -2 ]
[ -2 2 ] [ -2 2 ]
Multiply matrices:
A² = [ (2⋅2) + (-2⋅-2) (2⋅-2) + (-2⋅2) ]
[ (-2⋅2) + (2⋅-2) (-2⋅-2) + (2⋅2) ]
= [ 4 + 4 -4 - 4 ]
[ -4 - 4 4 + 4 ]
= [ 8 -8 ]
[ -8 8 ]
Now compute pA:
pA = p ⋅ [ 2 -2 ]
[ -2 2 ]
= [ 2p -2p ]
[ -2p 2p ]
Equating both matrices:
[ 8 -8 ] = [ 2p -2p ]
[ -8 8 ] [ -2p 2p ]
⇒ 2p = 8 ⇒ p = 4