Simplify the Matrix

Problem

[[0,1,0,2],[1,−1,−2,2],[0,1,3,−3],[−2,2,3,−2]]+[[0,−1,0,−2],[−1,1,2,−2],[0,−1,−3,3],[2,−2,−3,2]]+[[1,3,−2,−1],[−3,−12,10,6],[0,−2,2,1],[−2,−6,5,3]]+[[1,3,−2,−1],[−3,−12,10,6],[0,−2,2,1],[−2,−6,5,3]]+[[0,1,0,2],[1,−1,−2,2],[0,1,3,−3],[−2,2,3,−2]]

Solution

1. Identify the matrices to be added. Let the first matrix be A the second be B and the third be C The expression is A+B+C+C+A which simplifies to 2*A+B+2*C

2. Calculate the sum of the first two matrices A+B

[[0,1,0,2],[1,−1,−2,2],[0,1,3,−3],[−2,2,3,−2]]+[[0,−1,0,−2],[−1,1,2,−2],[0,−1,−3,3],[2,−2,−3,2]]=[[0,0,0,0],[0,0,0,0],[0,0,0,0],[0,0,0,0]]

3. Calculate the sum of the remaining matrices C+C+A or 2*C+A Since A+B resulted in the zero matrix, the total sum is simply 2*C+A

4. Multiply matrix C by 2.

2*[[1,3,−2,−1],[−3,−12,10,6],[0,−2,2,1],[−2,−6,5,3]]=[[2,6,−4,−2],[−6,−24,20,12],[0,−4,4,2],[−4,−12,10,6]]

5. Add the result to matrix A to find the final sum.

[[2,6,−4,−2],[−6,−24,20,12],[0,−4,4,2],[−4,−12,10,6]]+[[0,1,0,2],[1,−1,−2,2],[0,1,3,−3],[−2,2,3,−2]]=[[2,7,−4,0],[−5,−25,18,14],[0,−3,7,−1],[−6,−10,13,4]]

Final Answer

[[0,1,0,2],[1,−1,−2,2],[0,1,3,−3],[−2,2,3,−2]]+[[0,−1,0,−2],[−1,1,2,−2],[0,−1,−3,3],[2,−2,−3,2]]+[[1,3,−2,−1],[−3,−12,10,6],[0,−2,2,1],[−2,−6,5,3]]+[[1,3,−2,−1],[−3,−12,10,6],[0,−2,2,1],[−2,−6,5,3]]+[[0,1,0,2],[1,−1,−2,2],[0,1,3,−3],[−2,2,3,−2]]=[[2,7,−4,0],[−5,−25,18,14],[0,−3,7,−1],[−6,−10,13,4]]

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