How to calculate the following determinant:
$\begin{vmatrix} a & b & b & ... & b \\ b & a & b & ... & b \\ . & . & . & . ... & .\\ b & b & b & ... & a\\ \end{vmatrix}$
I don't have idea from where to start. Any help is welcome. Thanks in advance.
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You can calculate it with eigenvalues and their multiplicities. Think about the eigenvalues of the matrix $J$ with all $1$s. Then this matrix is just $bJ + (a - b)I$, which is a polynomial of $J$. – Theo Bendit Mar 07 '21 at 16:04
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You can try using elementary row operations to diagonalize the matrix since the determinant of a diagonal matrix is just the product of the diagonal entries. But remember, elementary row operations may change the determinant, so keep track of the operations you used. – JLinsta Mar 07 '21 at 16:19