Simplify the Matrix [[2,3],[4,1]]^4

Problem

[[2,3],[4,1]]^4

Solution

1. Define the matrix A as the given 2×2 matrix.

A=[[2,3],[4,1]]

2. Calculate A^2 by performing matrix multiplication A×A

A^2=[[2,3],[4,1]]*[[2,3],[4,1]]

A^2=[[(2)*(2)+(3)*(4),(2)*(3)+(3)*(1)],[(4)*(2)+(1)*(4),(4)*(3)+(1)*(1)]]

A^2=[[16,9],[12,13]]

3. Calculate A^4 by squaring the result of A^2 such that A^4=A^2×A^2

A^4=[[16,9],[12,13]]*[[16,9],[12,13]]

A^4=[[(16)*(16)+(9)*(12),(16)*(9)+(9)*(13)],[(12)*(16)+(13)*(12),(12)*(9)+(13)*(13)]]

4. Simplify the arithmetic for each entry in the resulting matrix.

A^4=[[256+108,144+117],[192+156,108+169]]

A^4=[[364,261],[348,277]]

Final Answer

[[2,3],[4,1]]^4=[[364,261],[348,277]]

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