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

Problem

[[1,−2],[3,−4]]^3

Solution

1. Identify the matrix A and the operation to be performed, which is matrix exponentiation A^3=A⋅A⋅A

A=[[1,−2],[3,−4]]

2. Calculate the square of the matrix A^2 by multiplying A by itself.

A^2=[[1,−2],[3,−4]]*[[1,−2],[3,−4]]

3. Compute the individual entries of A^2 using the dot product of rows and columns.

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

A^2=[[1−6,−2+8],[3−12,−6+16]]

A^2=[[−5,6],[−9,10]]

4. Calculate the cube of the matrix A^3 by multiplying A^2 by A

A^3=A^2⋅A=[[−5,6],[−9,10]]*[[1,−2],[3,−4]]

5. Compute the individual entries of A^3 using the dot product of rows and columns.

A^3=[[(−5)*(1)+(6)*(3),(−5)*(−2)+(6)*(−4)],[(−9)*(1)+(10)*(3),(−9)*(−2)+(10)*(−4)]]

A^3=[[−5+18,10−24],[−9+30,18−40]]

A^3=[[13,−14],[21,−22]]

Final Answer

[[1,−2],[3,−4]]^3=[[13,−14],[21,−22]]

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