Which is correct and why?

Question

Assume that there are two lower triangular matrices A and B of size n*n. If matrix A and transpose of B are fit into a rectangular matrix C of size n*(n+1), then
a) B[i,j]=C[i,j+1]
b)B[i,j]=C[j+1,i]
c)B[i,j]=C[j, i+1]

Answer 1

By the very definition of transpose(say T) of a matrix M, we have M[i, j] = T[j, i]. But here since we are merging matrix A with transpose of B, the diagonal elements of A and B will have to be preserved in C, therefore,

A[i, j] = C[j, i]

B[i, j] = C[j, i+1]

Here B[i, j] cannot be equal to C[j, i] because presence of elements of A at the diagonal of C shifts the elements of B by 1 column.

 

I would like to know a formal explanation though.

Comments

According to me, the B[i,j]=C[i,j+1] holds true here. and b) and c) are not. What you say?

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Do you mean,

$B^{T} [i, j] = C[i, j + 1]$?

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oops.