Simplify the Matrix [[1+i,1-i],[i,1]]*[[1-i,1+i],[-i,2]]

Problem

[[1+i,1−i],[i,1]]*[[1−i,1+i],[−i,2]]

Solution

1. Identify the matrix multiplication rule for 2×2 matrices, where the element in row i and column j is the dot product of the ith row of the first matrix and the jth column of the second matrix.

2. Calculate the top-left element by multiplying the first row of the first matrix by the first column of the second matrix.

(1+i)*(1−i)+(1−i)*(−i)

1−i2−i+i2

1−(−1)−i+(−1)

1+1−i−1=1−i

3. Calculate the top-right element by multiplying the first row of the first matrix by the second column of the second matrix.

(1+i)*(1+i)+(1−i)*(2)

1+2*i+i2+2−2*i

1+2*i−1+2−2*i=2

4. Calculate the bottom-left element by multiplying the second row of the first matrix by the first column of the second matrix.

(i)*(1−i)+(1)*(−i)

i−i2−i

i−(−1)−i=1

5. Calculate the bottom-right element by multiplying the second row of the first matrix by the second column of the second matrix.

(i)*(1+i)+(1)*(2)

i+i2+2

i−1+2=1+i

6. Assemble the resulting matrix using the calculated components.

Final Answer

[[1+i,1−i],[i,1]]*[[1−i,1+i],[−i,2]]=[[1−i,2],[1,1+i]]

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