Simplify the Matrix [[2,0],[9,4]]^3

Problem

[[2,0],[9,4]]3

Solution

1. Identify the matrix A and the operation to be performed, which is calculating A3=A⋅A⋅A

A=[[2,0],[9,4]]

2. Calculate A2 by multiplying the matrix by itself using the row-by-column rule.

A2=[[2,0],[9,4]]*[[2,0],[9,4]]

A2=[[(2)*(2)+(0)*(9),(2)*(0)+(0)*(4)],[(9)*(2)+(4)*(9),(9)*(0)+(4)*(4)]]

A2=[[4,0],[54,16]]

3. Calculate A3 by multiplying A2 by A

A3=A2⋅A=[[4,0],[54,16]]*[[2,0],[9,4]]

A3=[[(4)*(2)+(0)*(9),(4)*(0)+(0)*(4)],[(54)*(2)+(16)*(9),(54)*(0)+(16)*(4)]]

A3=[[8,0],[108+144,64]]

A3=[[8,0],[252,64]]

Final Answer

[[2,0],[9,4]]3=[[8,0],[252,64]]

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