Transpose [[1,1],[0,1]]^n

Problem

([[1,1],[0,1]]n)T

Solution

1. Identify the matrix power. The matrix A=[[1,1],[0,1]] is a standard shear matrix.

2. Compute the power An By observing the pattern of matrix multiplication:

A1=[[1,1],[0,1]]

A2=[[1,1],[0,1]]*[[1,1],[0,1]]=[[1,2],[0,1]]

A3=[[1,2],[0,1]]*[[1,1],[0,1]]=[[1,3],[0,1]]

3. Generalize the result for any integer n The nth power of the matrix is:

[[1,1],[0,1]]n=[[1,n],[0,1]]

4. Apply the transpose operation. The transpose of a matrix swaps its rows and columns:

([1,n],[0,1])T=[[1,0],[n,1]]

Final Answer

([[1,1],[0,1]]n)T=[[1,0],[n,1]]

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