Calculate a determinant

Jun 2019
493
262
USA
How could i calculate determinant this matrix. if it doesn't have exact dimension
The central diagonal {(i,i)} has the product \(\displaystyle (a+b)^N\). This is a "positive" diagonal. For a 3x3 or higher, all other diagonals have at least one zero in them. (Note, the "1"s diagonal has the zero from (1,N) and the "ab"s diagonal has the zero from (N,1). All the down-and-to-the-left diagonals also have at least one zero.) The determinant should be \(\displaystyle (a+b)^N \forall N \geq 3\), where N is the size of the matrix. Try it for 3x3 and 4x4 to test it. Happens to work for 1x1, also. 2x2 would be a special case.
 
Jun 2019
493
262
USA
Scratch the above answer. It's wrong, and for some reason I can't edit it.

Solved it by example (up to 5x5):
\(\displaystyle |2 \times 2| = a^2 + ab + b^2\)
\(\displaystyle |3 \times 3| = a^3 + a^2b + ab^2 + b^3\)
\(\displaystyle |N \times N| = \sum_{i=0}^{N} a^ib^{N-i}\)
 
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