Multiply the Matrices

Problem

[[1,0,0,0],[0,1,0,0],[2,−4,1,0],[−5,−3,−9,1]]*[[−3,2,0,5],[0,−4,0,3],[0,0,7,−1],[0,0,0,−9]]

Solution

1. Identify the dimensions of the matrices. Both are 4×4 matrices, so the resulting product will also be a 4×4 matrix.

2. Calculate the first row of the product. Since the first row of the left matrix is [1,0,0,0] it acts as an identity operation on the first row of the right matrix.

(R_1)=[1*(−3)+0*(0)+0*(0)+0*(0),1*(2)+0*(−4)+0*(0)+0*(0),1*(0)+0*(0)+0*(7)+0*(0),1*(5)+0*(3)+0*(−1)+0*(−9)]

(R_1)=[−3,2,0,5]

3. Calculate the second row of the product. Since the second row of the left matrix is [0,1,0,0] it returns the second row of the right matrix.

(R_2)=[0*(−3)+1*(0)+0*(0)+0*(0),0*(2)+1*(−4)+0*(0)+0*(0),0*(0)+1*(0)+0*(7)+0*(0),0*(5)+1*(3)+0*(−1)+0*(−9)]

(R_2)=[0,−4,0,3]

4. Calculate the third row of the product by taking the dot product of the third row of the left matrix with each column of the right matrix.

(R_3,1)=2*(−3)+(−4)*(0)+1*(0)+0*(0)=−6

(R_3,2)=2*(2)+(−4)*(−4)+1*(0)+0*(0)=4+16=20

(R_3,3)=2*(0)+(−4)*(0)+1*(7)+0*(0)=7

(R_3,4)=2*(5)+(−4)*(3)+1*(−1)+0*(−9)=10−12−1=−3

5. Calculate the fourth row of the product by taking the dot product of the fourth row of the left matrix with each column of the right matrix.

(R_4,1)=−5*(−3)+(−3)*(0)+(−9)*(0)+1*(0)=15

(R_4,2)=−5*(2)+(−3)*(−4)+(−9)*(0)+1*(0)=−10+12=2

(R_4,3)=−5*(0)+(−3)*(0)+(−9)*(7)+1*(0)=−63

(R_4,4)=−5*(5)+(−3)*(3)+(−9)*(−1)+1*(−9)=−25−9+9−9=−34

Final Answer

[[1,0,0,0],[0,1,0,0],[2,−4,1,0],[−5,−3,−9,1]]*[[−3,2,0,5],[0,−4,0,3],[0,0,7,−1],[0,0,0,−9]]=[[−3,2,0,5],[0,−4,0,3],[−6,20,7,−3],[15,2,−63,−34]]

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