Simplify the Matrix [[I-BA,B],[2A-ABA,AB-I]]^2

Problem

[[I−B*A,B],[2*A−A*B*A,A*B−I]]2

Solution

1. Identify the matrix M to be squared.

M=[[I−B*A,B],[2*A−A*B*A,A*B−I]]

2. Apply the formula for matrix multiplication M2=M⋅M

M2=[[I−B*A,B],[2*A−A*B*A,A*B−I]]*[[I−B*A,B],[2*A−A*B*A,A*B−I]]

3. Calculate the top-left entry (1,1) of the resulting matrix.

(I−B*A)*(I−B*A)+B*(2*A−A*B*A)

I−2*B*A+B*A*B*A+2*B*A−B*A*B*A

I

4. Calculate the top-right entry (1,2) of the resulting matrix.

(I−B*A)*B+B*(A*B−I)

B−B*A*B+B*A*B−B

0

5. Calculate the bottom-left entry (2,1) of the resulting matrix.

(2*A−A*B*A)*(I−B*A)+(A*B−I)*(2*A−A*B*A)

2*A−2*A*B*A−A*B*A+A*B*A*B*A+2*A*B*A−A*B*A*B*A−2*A+A*B*A

0

6. Calculate the bottom-right entry (2,2) of the resulting matrix.

(2*A−A*B*A)*B+(A*B−I)*(A*B−I)

2*A*B−A*B*A*B+A*B*A*B−2*A*B+I

I

7. Combine the calculated entries into the final matrix.

M2=[[I,0],[0,I]]

Final Answer

[[I−B*A,B],[2*A−A*B*A,A*B−I]]2=[[I,0],[0,I]]

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