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Bruno J. · #6 June 16th, 2009, 11:15 PM

Let $I_{(j)}$ be the identity matrix with row j multiplied by 2. Then $I_{(j)}A$ is A with row j multiplied by 2, and $AI_{(j)}$ is A with column j multiplied by 2. From

$I_{(j)}A=AI_{(j)}$

we conclude that A has a diagonal entry and zeroes everywhere else on the jth row and the jth column. In this fashion we conclude that A is diagonal.

It is then easy to show that the diagonal entries are all equal.