(X+Y)4 =
Expressions for Xn, Yn and (X+Y)n
Xn = Yn = (X+Y)n =
n > 0,
let: W = any 2x2 matrix,
W-n = → It is not possible to divide an integer by a matrix, n < 0 does not exist
n≠0
For any matrix where n=0 Wn = I W0 =
A=aX → a
B=bX → b
Let: a = 3 and b = 4
A2 =
A3 =
A4 =
B2 =
B3 =
B4 =
Therefore:
(A+B) =
(A+B)2 =
(A+B)3 =
(A+B)4 =
Expressions for An, Bn and (A+B)n
An = Bn =
(A+B)n =
n > 0,
let: W = any 22 matrix,
W-n = → It is not possible to divide an integer by a matrix, n < 0 does not exist
n≠0
For any matrix where n=0 Wn = I W0 =
Let: M = , M = A+B and M2 = A2+B2
A = aX = B = bY =
A+B =
Therefore: M = A+B
M= (A+B) =
An = A2 = Bn = B2 =
A2+B2 =
Therefore: M2 = A2+B2
A= aX = , 2a2X =
A2 =
B= bY = ,
2b2Y=
B2 = : M2 in terms of aX and bY = M2 = 2a2X + 2b2Y
M2 = 2a2X + 2b2Y = a(2a)2-1X + b(2b)2-1Y
Mn in terms of aX and bY
M = (A+B) =
Mn = a(2a)n-1X + b(2b)n-1Y
a(2a)n-1X =
b(2b)n-1X =
Mn =
M2 = a(2a)2-1X + b(2b)2-1Y
Therefore the general statement for Mn: Mn = a(2a)n-1X + b(2b)n-1Y
Testing the statement Mn = a(2a)n-1X + b(2b)n-1Y with real numbers
Let: a=2
b=3
n=4
M =
M4 =
=
M4 = a(2a)3X + b(2b)3Y
General statement is valid
Scope and limitations:
The preceding calculations determined general expressions for the defined matrices An, Bn, (A+B)n and Mn. The mentioned matrices however, were all 2 square matrices. Although the determined expressions are valid for 2 matrices, they may not be valid for square matrices with more columns and rows, or non-square matrices. The value of n must also be greater than zero. It cannot equal zero because any matrix raised to the power of zero would equal I the identity matrix, and it cannot be less than zero, because that would be the same as one divided by the matrix raised to the power of n. Since it is not possible to divide an integer by a matrix, it is not possible to raise any matrix to a negative exponent and this n must be greater than zero. Therefore the expressions determined in this portfolio are valid for 2matrices, where the value of n is greater than zero
