Simplify the Matrix

Problem

[[0,−1,−1],[−1,0,−7/2],[1,1,1]]*[[3,0,0],[0,−17,0],[0,0,18]]*[[1,0,1],[−5/7,2/7,1],[−2/7,−2/7,−2/7]]

Solution

1. Multiply the first matrix by the second (diagonal) matrix. Multiplying a matrix A by a diagonal matrix D on the right scales the columns of A but here we multiply A on the left of D which scales the columns of the first matrix by the diagonal entries of the second.

[[0,−1,−1],[−1,0,−7/2],[1,1,1]]*[[3,0,0],[0,−17,0],[0,0,18]]=[[0*(3),−1*(−17),−1*(18)],[−1*(3),0*(−17),−7/2*(18)],[1*(3),1*(−17),1*(18)]]

[[0,17,−18],[−3,0,−63],[3,−17,18]]

2. Multiply the resulting matrix by the third matrix. We perform standard matrix multiplication R=A*B where (R_i*j)=(∑_^)((A_i*k))*(B_k*j)

(R_11)=(0)*(1)+(17)*(−5/7)+(−18)*(−2/7)=0−85/7+36/7=−49/7=−7

(R_12)=(0)*(0)+(17)*(2/7)+(−18)*(−2/7)=0+34/7+36/7=70/7=10

(R_13)=(0)*(1)+(17)*(1)+(−18)*(−2/7)=0+17+36/7=(119+36)/7=155/7

3. Calculate the second row of the product.

(R_21)=(−3)*(1)+(0)*(−5/7)+(−63)*(−2/7)=−3+0+18=15

(R_22)=(−3)*(0)+(0)*(2/7)+(−63)*(−2/7)=0+0+18=18

(R_23)=(−3)*(1)+(0)*(1)+(−63)*(−2/7)=−3+0+18=15

4. Calculate the third row of the product.

(R_31)=(3)*(1)+(−17)*(−5/7)+(18)*(−2/7)=3+85/7−36/7=3+49/7=3+7=10

(R_32)=(3)*(0)+(−17)*(2/7)+(18)*(−2/7)=0−34/7−36/7=−70/7=−10

(R_33)=(3)*(1)+(−17)*(1)+(18)*(−2/7)=3−17−36/7=−14−36/7=(−98−36)/7=−134/7

Final Answer

[[0,−1,−1],[−1,0,−7/2],[1,1,1]]*[[3,0,0],[0,−17,0],[0,0,18]]*[[1,0,1],[−5/7,2/7,1],[−2/7,−2/7,−2/7]]=[[−7,10,155/7],[15,18,15],[10,−10,−134/7]]

Want more problems? Check here!