Simplify the Matrix

Problem

[[0.1,0.0,0.1,0.1,0.2],[0.1,0.2,−0.1,0.1,−0.1],[0.1,−0.1,0.1,−0.2,0.1],[0.1,0.2,0.1,0.1,0.1],[0.1,0.0,0.0,0.0,0.0]]*[[0,0,0,0,10],[0,5,10/3,5/3,−10],[−10,−15,−20/3,35/3,20],[0,−5,−20/3,5/3,10],[10,10,20/3,−20/3,−20]]

Solution

1. Convert the decimal matrix to a fraction matrix to simplify calculations, noting that 0.1=1/10 and 0.2=2/10

A=1/10*[[1,0,1,1,2],[1,2,−1,1,−1],[1,−1,1,−2,1],[1,2,1,1,1],[1,0,0,0,0]]

2. Factor out a common term from the second matrix to make multiplication easier.

B=5/3*[[0,0,0,0,6],[0,3,2,1,−6],[−6,−9,−4,7,12],[0,−3,−4,1,6],[6,6,4,−4,−12]]

3. Multiply the scalar constants outside the matrices.

1/10⋅5/3=5/30=1/6

4. Perform the matrix multiplication C=A′*B′ where A′ and B′ are the integer matrices.

(C_11)=(1)*(0)+(0)*(0)+(1)*(−6)+(1)*(0)+(2)*(6)=6

(C_12)=(1)*(0)+(0)*(3)+(1)*(−9)+(1)*(−3)+(2)*(6)=0

(C_13)=(1)*(0)+(0)*(2)+(1)*(−4)+(1)*(−4)+(2)*(4)=0

(C_14)=(1)*(0)+(0)*(1)+(1)*(7)+(1)*(1)+(2)*(−4)=0

(C_15)=(1)*(6)+(0)*(−6)+(1)*(12)+(1)*(6)+(2)*(−12)=0

5. Continue calculating the remaining rows of the product matrix.

(C_21)=(1)*(0)+(2)*(0)+(−1)*(−6)+(1)*(0)+(−1)*(6)=0

(C_22)=(1)*(0)+(2)*(3)+(−1)*(−9)+(1)*(−3)+(−1)*(6)=6

(C_33)=(1)*(0)+(−1)*(2)+(1)*(−4)+(−2)*(−4)+(1)*(4)=6

(C_44)=(1)*(0)+(2)*(1)+(1)*(7)+(1)*(1)+(1)*(−4)=6

(C_55)=(1)*(6)+(0)*(−6)+(0)*(12)+(0)*(6)+(0)*(−12)=6

6. Observe that the resulting integer matrix is 6*I where I is the identity matrix.

C=[[6,0,0,0,0],[0,6,0,0,0],[0,0,6,0,0],[0,0,0,6,0],[0,0,0,0,6]]

7. Multiply by the scalar 1/6 to find the final result.

1/6*[[6,0,0,0,0],[0,6,0,0,0],[0,0,6,0,0],[0,0,0,6,0],[0,0,0,0,6]]=[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]

Final Answer

[[0.1,0.0,0.1,0.1,0.2],[0.1,0.2,−0.1,0.1,−0.1],[0.1,−0.1,0.1,−0.2,0.1],[0.1,0.2,0.1,0.1,0.1],[0.1,0.0,0.0,0.0,0.0]]*[[0,0,0,0,10],[0,5,10/3,5/3,−10],[−10,−15,−20/3,35/3,20],[0,−5,−20/3,5/3,10],[10,10,20/3,−20/3,−20]]=[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]

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