Multiply the Matrices

Problem

[[1,1/115,−1/115],[3/53,−206/6095,13/6095],[−1/53,−2/6095,296/6095]]*[[296,86,300,162,225,459,487,77,763],[255,140,496,130,388,253,419,113,533],[405,405,367,105,646,736,748,216,856]]

Solution

1. Identify the dimensions of the matrices. The first matrix is 3×3 and the second is 3×9 so the resulting matrix will be 3×9

2. Find a common denominator for the first matrix to simplify calculations. Note that 6095=53×115

3. Rewrite the first matrix with a common denominator of 6095

[[6095/6095,53/6095,−53/6095],[345/6095,−206/6095,13/6095],[−115/6095,−2/6095,296/6095]]=1/6095*[[6095,53,−53],[345,−206,13],[−115,−2,296]]

4. Calculate the first row of the product by multiplying the first row of the left matrix by each column of the right matrix.

(R_1)*(C_1)=1/6095*(6095×296+53×255−53×405)=(1804120+13515−21465)/6095=1796170/6095=294.6956...

Wait, let's re-examine the first matrix. 1+1/115*(255)−1/115*(405)=1+(255−405)/115=1−150/115=1−30/23=−7/23
Actually, let's perform the standard row-by-column multiplication (c_i*j)=(∑_^)((a_i*k))*(b_k*j)

5. Compute the first row elements:

(c_11)=1*(296)+1/115*(255)−1/115*(405)=296−150/115=296−30/23=6778/23

(c_12)=1*(86)+1/115*(140)−1/115*(405)=86−265/115=86−53/23=1925/23

(c_13)=1*(300)+1/115*(496)−1/115*(367)=300+129/115

(c_14)=1*(162)+1/115*(130)−1/115*(105)=162+25/115=162+5/23=3731/23

(c_15)=1*(225)+1/115*(388)−1/115*(646)=225−258/115

(c_16)=1*(459)+1/115*(253)−1/115*(736)=459−483/115=459−21/5=2274/5

(c_17)=1*(487)+1/115*(419)−1/115*(748)=487−329/115

(c_18)=1*(77)+1/115*(113)−1/115*(216)=77−103/115

(c_19)=1*(763)+1/115*(533)−1/115*(856)=763−323/115

6. Compute the second row elements:

(c_21)=3/53*(296)−206/6095*(255)+13/6095*(405)=(102120−52530+5265)/6095=54855/6095=9

(c_22)=3/53*(86)−206/6095*(140)+13/6095*(405)=(29670−28840+5265)/6095=6095/6095=1

(c_23)=3/53*(300)−206/6095*(496)+13/6095*(367)=(103500−102176+4771)/6095=6095/6095=1

(c_24)=3/53*(162)−206/6095*(130)+13/6095*(105)=(55890−26780+1365)/6095=30475/6095=5

(c_25)=3/53*(225)−206/6095*(388)+13/6095*(646)=(77625−79928+8398)/6095=6095/6095=1

(c_26)=3/53*(459)−206/6095*(253)+13/6095*(736)=(158355−52118+9568)/6095=115805/6095=19

(c_27)=3/53*(487)−206/6095*(419)+13/6095*(748)=(168015−86314+9724)/6095=91425/6095=15

(c_28)=3/53*(77)−206/6095*(113)+13/6095*(216)=(26565−23278+2808)/6095=6095/6095=1

(c_29)=3/53*(763)−206/6095*(533)+13/6095*(856)=(263235−109798+11128)/6095=164565/6095=27

7. Compute the third row elements:

(c_31)=−1/53*(296)−2/6095*(255)+296/6095*(405)=(−34040−510+119880)/6095=85330/6095=14

(c_32)=−1/53*(86)−2/6095*(140)+296/6095*(405)=(−9890−280+119880)/6095=109710/6095=18

(c_33)=−1/53*(300)−2/6095*(496)+296/6095*(367)=(−34500−992+108632)/6095=73140/6095=12

(c_34)=−1/53*(162)−2/6095*(130)+296/6095*(105)=(−18630−260+31080)/6095=12190/6095=2

(c_35)=−1/53*(225)−2/6095*(388)+296/6095*(646)=(−25875−776+191216)/6095=164565/6095=27

(c_36)=−1/53*(459)−2/6095*(253)+296/6095*(736)=(−52785−506+217856)/6095=164565/6095=27

(c_37)=−1/53*(487)−2/6095*(419)+296/6095*(748)=(−56005−838+221408)/6095=164565/6095=27

(c_38)=−1/53*(77)−2/6095*(113)+296/6095*(216)=(−8855−226+63936)/6095=54855/6095=9

(c_39)=−1/53*(763)−2/6095*(533)+296/6095*(856)=(−87745−1066+253376)/6095=164565/6095=27

8. Re-evaluate the first row. Given the integer patterns in rows 2 and 3, it is likely the first row was intended to be simpler. Let's re-calculate (c_11) carefully: 296+(255−405)/115=296−150/115=296−30/23=(6808−30)/23=6778/23 Since the task asks to multiply the matrices as given, we provide the exact values.

Final Answer

[[1,1/115,−1/115],[3/53,−206/6095,13/6095],[−1/53,−2/6095,296/6095]]*[[296,86,300,162,225,459,487,77,763],[255,140,496,130,388,253,419,113,533],[405,405,367,105,646,736,748,216,856]]=[[6778/23,1925/23,34629/115,3731/23,25617/115,2274/5,55676/115,8752/115,87422/115],[9,1,1,5,1,19,15,1,27],[14,18,12,2,27,27,27,9,27]]

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