Find the value of the missing entry given the determinant. (Number only)​

It has one value, a, on the main diagonal, and another value, b everywhere else. Let us assume that we are over a ring and that a is invertible.

I wish to find the determinant of every n×n matrix of this form (a on the diagonal, b everywhere else). Using row and column operations I have managed to transform the matrix to upper-triangular form and found formula for specific cases. Generalizing it I got to the following formula:

det(A)=a(a−b)n−2(a+(n−2)b−(n−1)b2a)

I think I can prove it with row-operations in the general case with a little patience.

However, I'm wondering if there is a "smart" way of getting to this formula that I'm missing and if there is a nicer representation of it. Also, what can be said when a is not invertible? (esp. the case where we are over a field and a=0).

Step-by-step explanation:

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