Multiply the Matrices

Problem

[[i+1,2,1,−i],[1,i,2*i,2]]*[[12,33,4,3],[i,4,65,4],[1,i,i,2],[1,1,0,i]]

Solution

1. Identify the dimensions of the matrices. The first matrix is 2×4 and the second is 4×4 so the resulting matrix will be 2×4

2. Calculate the first row, first column entry (c_11) by taking the dot product of the first row of the first matrix and the first column of the second matrix.

(c_11)=(i+1)*(12)+(2)*(i)+(1)*(1)+(−i)*(1)

(c_11)=12*i+12+2*i+1−i=13+13*i

3. Calculate the first row, second column entry (c_12)

(c_12)=(i+1)*(33)+(2)*(4)+(1)*(i)+(−i)*(1)

(c_12)=33*i+33+8+i−i=41+33*i

4. Calculate the first row, third column entry (c_13)

(c_13)=(i+1)*(4)+(2)*(65)+(1)*(i)+(−i)*(0)

(c_13)=4*i+4+130+i=134+5*i

5. Calculate the first row, fourth column entry (c_14)

(c_14)=(i+1)*(3)+(2)*(4)+(1)*(2)+(−i)*(i)

(c_14)=3*i+3+8+2−i2=3*i+13−(−1)=14+3*i

6. Calculate the second row, first column entry (c_21)

(c_21)=(1)*(12)+(i)*(i)+(2*i)*(1)+(2)*(1)

(c_21)=12+i2+2*i+2=12−1+2*i+2=13+2*i

7. Calculate the second row, second column entry (c_22)

(c_22)=(1)*(33)+(i)*(4)+(2*i)*(i)+(2)*(1)

(c_22)=33+4*i+2*i2+2=33+4*i−2+2=33+4*i

8. Calculate the second row, third column entry (c_23)

(c_23)=(1)*(4)+(i)*(65)+(2*i)*(i)+(2)*(0)

(c_23)=4+65*i+2*i2=4+65*i−2=2+65*i

9. Calculate the second row, fourth column entry (c_24)

(c_24)=(1)*(3)+(i)*(4)+(2*i)*(2)+(2)*(i)

(c_24)=3+4*i+4*i+2*i=3+10*i

Final Answer

[[i+1,2,1,−i],[1,i,2*i,2]]*[[12,33,4,3],[i,4,65,4],[1,i,i,2],[1,1,0,i]]=[[13+13*i,41+33*i,134+5*i,14+3*i],[13+2*i,33+4*i,2+65*i,3+10*i]]

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