Find the Variables

Problem

A×B,A=[[2,0,1],[3,0,0],[5,1,1]],B=[[1,0,1],[1,2,1],[1,1,0]]

Solution

1. Identify the dimensions of the matrices. Both A and B are 3×3 matrices, so their product C=A×B will also be a 3×3 matrix.

2. Calculate the elements of the first row of the product matrix by multiplying the first row of A by each column of B

(c_11)=(2)*(1)+(0)*(1)+(1)*(1)=3

(c_12)=(2)*(0)+(0)*(2)+(1)*(1)=1

(c_13)=(2)*(1)+(0)*(1)+(1)*(0)=2

3. Calculate the elements of the second row of the product matrix by multiplying the second row of A by each column of B

(c_21)=(3)*(1)+(0)*(1)+(0)*(1)=3

(c_22)=(3)*(0)+(0)*(2)+(0)*(1)=0

(c_23)=(3)*(1)+(0)*(1)+(0)*(0)=3

4. Calculate the elements of the third row of the product matrix by multiplying the third row of A by each column of B

(c_31)=(5)*(1)+(1)*(1)+(1)*(1)=7

(c_32)=(5)*(0)+(1)*(2)+(1)*(1)=3

(c_33)=(5)*(1)+(1)*(1)+(1)*(0)=6

5. Assemble the resulting values into the final product matrix.

Final Answer

[[2,0,1],[3,0,0],[5,1,1]]×[[1,0,1],[1,2,1],[1,1,0]]=[[3,1,2],[3,0,3],[7,3,6]]

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