Multiply the Matrices

Problem

[[7/24,1/12,1/12,1/24],[1/12,7/24,1/24,1/12],[1/12,1/24,7/24,1/12],[1/24,1/12,1/12,7/24]]*[[20],[50],[15],[50]]

Solution

1. Convert all fractions to a common denominator of 24 to simplify the arithmetic.

[[7/24,2/24,2/24,1/24],[2/24,7/24,1/24,2/24],[2/24,1/24,7/24,2/24],[1/24,2/24,2/24,7/24]]*[[20],[50],[15],[50]]

2. Calculate the first row entry by multiplying the first row of the matrix by the column vector.

(7*(20)+2*(50)+2*(15)+1*(50))/24=(140+100+30+50)/24=320/24=40/3

3. Calculate the second row entry by multiplying the second row of the matrix by the column vector.

(2*(20)+7*(50)+1*(15)+2*(50))/24=(40+350+15+100)/24=505/24

4. Calculate the third row entry by multiplying the third row of the matrix by the column vector.

(2*(20)+1*(50)+7*(15)+2*(50))/24=(40+50+105+100)/24=295/24

5. Calculate the fourth row entry by multiplying the fourth row of the matrix by the column vector.

(1*(20)+2*(50)+2*(15)+7*(50))/24=(20+100+30+350)/24=500/24=125/6

Final Answer

[[7/24,1/12,1/12,1/24],[1/12,7/24,1/24,1/12],[1/12,1/24,7/24,1/12],[1/24,1/12,1/12,7/24]]*[[20],[50],[15],[50]]=[[40/3],[505/24],[295/24],[125/6]]

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