Multiply the Matrices

Problem

[[1+i,1/5,i+7],[1/2+√(,3)/2*i,−i,2+2*i]]*[[e(i*π)/4,−1],[−2*i,5+5*i],[i,3]]

Solution

1. Convert the exponential form e(i*π)/4 to rectangular form using Euler's formula e(i*θ)=cos(θ)+i*sin(θ)

cos(π/4)+i*sin(π/4)=√(,2)/2+√(,2)/2*i

2. Set up the matrix multiplication for the element in the first row, first column ((c_11) by multiplying the first row of the first matrix by the first column of the second matrix.

(c_11)=(1+i)*(√(,2)/2+√(,2)/2*i)+(1/5)*(−2*i)+(i+7)*(i)

(c_11)=√(,2)*i−2/5*i−1+7*i=−1+(√(,2)+33/5)*i

3. Calculate the element in the first row, second column ((c_12) by multiplying the first row of the first matrix by the second column of the second matrix.

(c_12)=(1+i)*(−1)+(1/5)*(5+5*i)+(i+7)*(3)

(c_12)=−1−i+1+i+3*i+21=21+3*i

4. Calculate the element in the second row, first column ((c_21) by multiplying the second row of the first matrix by the first column of the second matrix.

(c_21)=(1/2+√(,3)/2*i)*(√(,2)/2+√(,2)/2*i)+(−i)*(−2*i)+(2+2*i)*(i)

(c_21)=(√(,2)−√(,6))/4+(√(,2)+√(,6))/4*i−2+2*i−2=(√(,2)−√(,6)−16)/4+(√(,2)+√(,6)+8)/4*i

5. Calculate the element in the second row, second column ((c_22) by multiplying the second row of the first matrix by the second column of the second matrix.

(c_22)=(1/2+√(,3)/2*i)*(−1)+(−i)*(5+5*i)+(2+2*i)*(3)

(c_22)=−1/2−√(,3)/2*i−5*i+5+6+6*i=21/2+(2−√(,3))/2*i

Final Answer

[[1+i,1/5,i+7],[1/2+√(,3)/2*i,−i,2+2*i]]*[[e(i*π)/4,−1],[−2*i,5+5*i],[i,3]]=[[−1+(√(,2)+33/5)*i,21+3*i],[(√(,2)−√(,6)−16)/4+(√(,2)+√(,6)+8)/4*i,21/2+(2−√(,3))/2*i]]

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