Multiply the Matrices

Problem

[[1,t],[1/t,1]]*[[1,t],[1/t,1]]*[[1,t],[1/t,1]]*[[1,t],[1/t,1]]

Solution

1. Identify the matrix A and the goal to calculate A4

A=[[1,t],[1/t,1]]

2. Calculate the square of the matrix A2=A⋅A

A2=[[(1)*(1)+(t)*(1/t),(1)*(t)+(t)*(1)],[(1/t)*(1)+(1)*(1/t),(1/t)*(t)+(1)*(1)]]

3. Simplify the resulting entries of A2

A2=[[1+1,t+t],[1/t+1/t,1+1]]

A2=[[2,2*t],[2/t,2]]

4. Observe that A2=2*A This implies that for any power n An=2(n−1)*A

5. Calculate A4 by squaring A2

A4=A2⋅A2=[[2,2*t],[2/t,2]]*[[2,2*t],[2/t,2]]

6. Perform the final multiplication.

A4=[[(2)*(2)+(2*t)*(2/t),(2)*(2*t)+(2*t)*(2)],[(2/t)*(2)+(2)*(2/t),(2/t)*(2*t)+(2)*(2)]]

A4=[[4+4,4*t+4*t],[4/t+4/t,4+4]]

A4=[[8,8*t],[8/t,8]]

Final Answer

[[1,t],[1/t,1]]*[[1,t],[1/t,1]]*[[1,t],[1/t,1]]*[[1,t],[1/t,1]]=[[8,8*t],[8/t,8]]

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