Multiply the Matrices

Problem

[[1,8,0],[0,0,1],[0,1,0]]*[[4,0,−8],[0,5,0],[0,0,5]]

Solution

1. Identify the dimensions of the matrices. Both are 3×3 matrices, so the resulting product will also be a 3×3 matrix.

2. Calculate the elements of the first row by multiplying the first row of the left matrix by each column of the right matrix.

(R_1)*(C_1)=(1)*(4)+(8)*(0)+(0)*(0)=4

(R_1)*(C_2)=(1)*(0)+(8)*(5)+(0)*(0)=40

(R_1)*(C_3)=(1)*(−8)+(8)*(0)+(0)*(5)=−8

3. Calculate the elements of the second row by multiplying the second row of the left matrix by each column of the right matrix.

(R_2)*(C_1)=(0)*(4)+(0)*(0)+(1)*(0)=0

(R_2)*(C_2)=(0)*(0)+(0)*(5)+(1)*(0)=0

(R_2)*(C_3)=(0)*(−8)+(0)*(0)+(1)*(5)=5

4. Calculate the elements of the third row by multiplying the third row of the left matrix by each column of the right matrix.

(R_3)*(C_1)=(0)*(4)+(1)*(0)+(0)*(0)=0

(R_3)*(C_2)=(0)*(0)+(1)*(5)+(0)*(0)=5

(R_3)*(C_3)=(0)*(−8)+(1)*(0)+(0)*(5)=0

5. Assemble the results into the final product matrix.

Final Answer

[[1,8,0],[0,0,1],[0,1,0]]*[[4,0,−8],[0,5,0],[0,0,5]]=[[4,40,−8],[0,0,5],[0,5,0]]

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