3.求与下列矩阵可交换的矩阵:(2)(}1&0&00&2&00&0&3)

设 $ A = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{pmatrix} $，求与 $ A $ 可交换的矩阵 $ B $。 令 $ B = \begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{pmatrix} $，则 \[ AB = \begin{pmatrix} a_{11} & a_{12} & a_{13} \\ 2a_{21} & 2a_{22} & 2a_{23} \\ 3a_{31} & 3a_{32} & 3a_{33} \end{pmatrix}, \quad BA = \begin{pmatrix} a_{11} & 2a_{12} & 3a_{13} \\ a_{21} & 2a_{22} & 3a_{23} \\ a_{31} & 2a_{32} & 3a_{33} \end{pmatrix} \] 由 $ AB = BA $，得 \[ \begin{cases} a_{12} = 2a_{12} \Rightarrow a_{12} = 0 \\ a_{13} = 3a_{13} \Rightarrow a_{13} = 0 \\ 2a_{21} = a_{21} \Rightarrow a_{21} = 0 \\ 2a_{23} = 3a_{23} \Rightarrow a_{23} = 0 \\ 3a_{31} = a_{31} \Rightarrow a_{31} = 0 \\ 3a_{32} = 2a_{32} \Rightarrow a_{32} = 0 \end{cases} \] 其余元素无限制，故 $ B $ 为对角矩阵： \[ \boxed{\begin{pmatrix} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{pmatrix}} \] 其中 $ a, b, c $ 为任意实数。