Multiply the Matrices

Problem

[[0,1/115,−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],[405,405,367,105,646,736,748,216,85]]

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. Factor out common denominators to simplify calculations. Let the first matrix be A Note that 6095=53×115 We can write A as:

A=1/6095*[[0,53,−53],[345,−206,13],[−115,−2,296]]

3. Calculate the first row of the product by multiplying the first row of A by each column of the second matrix. For example, the first element is:

1/6095*(0×296+53×255−53×405)=(53*(255−405))/6095=(53*(−150))/6095=(−7950)/6095=−150/115=−30/23

4. Calculate the remaining elements using the standard matrix multiplication rule (C_i*j)=(∑_k=1^3)((A_i*k))*(B_k*j)

5. Simplify each resulting fraction to its lowest terms.

Final Answer

[[0,1/115,−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],[405,405,367,105,646,736,748,216,85]]=[[−30/23,−53/23,129/115,5/23,−258/115,−483/115,−329/115,−103/115,448/115],[103/115,13/115,1/5,107/115,13/23,13/5,17/5,1/23,456/115],[101/115,19/115,14/115,23/115,28/115,31/115,32/115,9/115,11/115]]

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