+68 Let A and B be the following matrices:
A rotates vectors by $\frac{\pi}{4}$ counterclockwise,
B is matrix \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}.
Calculate \[\mathbf{A}^{102} \mathbf{B}^{102} - \mathbf{B}^{102} \mathbf{A}^{102} \]
and enter it below.
+129895 A2 = [1/sqrt 2 -1/sqrt 2] * [ 1/sqrt 2 -1/sqrt2 ] = [ 0 -1 ]
[ 1/sqrt 2 1/sqrt 2] [ 1/sqrt 2 1/sqrt2] [ 1 0 ]
A4 = A2 * A2 = [ 0 -1 ] * [ 0 -1 ] = [ -1 0 ]
[ 1 0 ] [ 1 0 ] [ 0 -1]
A6 = A4 * A2 = [ -1 0 ] * [ 0 -1] = [ 0 1 ]
[ 0 -1 ] [ 1 0 ] [ -1 0]
A12 = (A6)2 = [ 0 1 ]
[ --1 0 ]
Note that (A6)n = [ 0 1 ] where n = 1,2,3,4.......
[ -1 0]
A102 = (A6)17 = [ 0 1 ]
[-1 0 ]
B2 = [ 1 1 ] * [ 1 1 ] = [ 1 2 ]
[ 0 1 ] [ 0 1] [ 0 1]
B4 = [ 1 2 ] * [ 1 2 ] = [1 4 ]
[ 0 1] [0 1 ] [ 0 1]
B6 = B4 * B2 = [ 1 4 ] * [ 1 2 ] = [ 1 6 ]
[ 0 1] [ 0 1] [ 0 1]
Note that B2n = [ 1 2n ] where n =1,2,3,4.......
[ 0 1]
B102 = B2*51 = [ 1 2*51 ] = [ 1 102 ]
[ 0 1 ] [ 0 1 ]
A102B102 = [ 0 1 ] * [ 1 102 ] = [ 0 1 ]
[ -1 0 ] [ 0 1 ] [ -1 -102 ]
B102 A102 = [ 1 102 ] * [ 0 1 ] = [ -102 1 ]
[ 0 1] [ -1 0 ] [ -1 0 ]
A102B102 - B102A102 = [ 102 0 ]
[ 0 -102 ]
