# 单位矩阵对矩阵的影响：左行右列，先近后远

## 一、题目

$$\boldsymbol{A}=\left[\begin{array}{lll} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{array}\right]$$

$$\boldsymbol{B}=\left[\begin{array}{ccc} a_{21} & a_{22} & a_{23} \\ a_{11} & a_{12} & a_{13} \\ a_{31}+a_{11} & a_{32}+a_{12} & a_{33}+a_{13} \end{array}\right]$$

$$\boldsymbol{P}_{1}=\left[\begin{array}{lll} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{array}\right]$$

$$\boldsymbol{P}_{2}=\left[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 1 & 0 & 1 \end{array}\right]$$

(A) $A \boldsymbol{P}_{1} \boldsymbol{P}_{2}=\boldsymbol{B}$

(B) $\boldsymbol{A} \boldsymbol{P}_{2} \boldsymbol{P}_{1}=\boldsymbol{B}$

(C) $\boldsymbol{P}_{1} \boldsymbol{P}_{2} \boldsymbol{A}=\boldsymbol{B}$

(D) $\boldsymbol{P}_{2} \boldsymbol{P}_{1} \boldsymbol{A}=\boldsymbol{B}$

## 二、解析

A 选项：

$$A P_{1} P_{2} \Rightarrow (A P_{1}) P_{2} \Rightarrow$$

B 选项：

$$A P_{2} P_{1} \Rightarrow (A P_{2}) P_{1} \Rightarrow$$

C 选项：

$$P_{1} P_{2} A \Rightarrow P_{1} (P_{2} A) \Rightarrow$$

D 选项：

$$P_{2} P_{1} A \Rightarrow P_{2} (P_{1} A) \Rightarrow$$