Determine all the matrix A 3x3 in R such as AB=BA for all B 3x3 in R (without determinants)
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Are you asking for ${ A | AB=BA \text{ for all } B }$? – copper.hat Jan 28 '20 at 03:14
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I'm sorry, yes, your correction is correct. – Alejandro Garcia Jan 28 '20 at 03:17
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Let $e_k$ be the vector of zeroes with one in the $k$th position.
Then $Ae_ie_j^T = e_i e_j^T A$ and so $e_a^T Ae_ie_j^T e_b = e_a^T e_i e_j^T A e_b$, or more simply $[A]_{ai} \delta_{jb} = [A]_{jb} \delta_{ai}$.
Setting $a=i, j=b$ gives $[A]_{ii} = [A]_{jj}$ and setting $a=i$ and $j \neq b$ gives $0=[A]_{jb}$.
Hence $A$ must be in the span of the identity.
copper.hat
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You found the identity matrix but if we consider the identity matrix and the zero matrix, can we find other Matrix B such as AB=BA? – Alejandro Garcia Jan 28 '20 at 03:41
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I didn't find the identity matrix, I found a matrix whose diagonal elements are all the same. Any matrix of the form $tI$ will do. – copper.hat Jan 28 '20 at 03:43