C (gcc), 79 77 bytes

• Saved two bytes thanks to Steadybox; only taking in one matrix dimension parameter as all matrices in this challenge are square.

j,i;f(A,n)int*A;{for(j=0;j<n*n;j++)printf("%d,",A[A[j]>A[i=j/n+j%n*n]?j:i]);} 

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Takes a flat integer array A and the matrix dimension n (as the matrix has to be square) as input. Outputs a flat integer array string representation to stdout.