2.1矩阵基础运算
加减：同型矩阵对应元素相加减
数乘：<span class="equation-text" contenteditable="false" data-index="0" data-equation="kA"><span></span><span></span></span>全部元素乘以k
矩阵乘法<span class="equation-text" contenteditable="false" data-index="0" data-equation="C=AB"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="(AB)_{ij}=\sum_{k=1}^n a_{ik}b_{kj}"><span></span><span></span></span>
A列数 = B行数
<span class="equation-text" contenteditable="false" data-index="0" data-equation="AB\neq BA"><span></span><span></span></span>，顺序不可交换
<span class="equation-text" contenteditable="false" data-index="0" data-equation="AB=0"><span></span><span></span></span>，推不出<span class="equation-text" contenteditable="false" data-index="0" data-equation="A=0"><span></span><span></span></span>或者<span class="equation-text" contenteditable="false" data-index="0" data-equation="B=0"><span></span><span></span></span>
📌2024计算题
<span class="equation-text" contenteditable="false" data-index="0" data-equation="A=\begin{pmatrix}1&1\\0&1\end{pmatrix},B=\begin{pmatrix}1&0\\2&1\end{pmatrix}"><span></span><span></span></span>
AB、BA分别计算，可以验证不等
2.2矩阵转置
<span class="equation-text" contenteditable="false" data-index="0" data-equation="(A^T)^T=A"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="(A+B)^T=A^T+B^T"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="(kA)^T=kA^T"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\boldsymbol{(AB)^T=B^T A^T}"><span></span><span></span></span>【逆序！高频考点】
2.3逆矩阵（重中之重）
定义：方阵<span class="equation-text" contenteditable="false" data-index="0" data-equation="AB=BA=E"><span></span><span></span></span>，<span class="equation-text" contenteditable="false" data-index="0" data-equation="B=A^{-1}"><span></span><span></span></span>
可逆充要条件：<span class="equation-text" contenteditable="false" data-index="0" data-equation="|A|\neq0"><span></span><span></span></span>（非奇异矩阵）
核心公式 <span class="equation-text" contenteditable="false" data-index="0" data-equation="A^{-1}=\dfrac{1}{|A|}A^*"><span></span><span></span></span>
二阶矩阵速记<span class="equation-text" contenteditable="false" data-index="0" data-equation="\begin{pmatrix}a&b\\c&d\end{pmatrix}^{-1}=\dfrac{1}{ad-bc}\begin{pmatrix}d&-b\\-c&a\end{pmatrix}"><span></span><span></span></span>
口诀：主对角线互换，副对角线变负号，整体除以行列式
运算性质
<span class="equation-text" contenteditable="false" data-index="0" data-equation="(kA)^{-1}=\dfrac1k A^{-1}"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\boldsymbol{(AB)^{-1}=B^{-1}A^{-1}}"><span></span><span></span></span>逆序
📌2024‑04真题
<span class="equation-text" contenteditable="false" data-index="0" data-equation="A=\begin{pmatrix}1&2\\0&1\end{pmatrix}"><span></span><span></span></span>求逆矩阵
<span class="equation-text" contenteditable="false" data-index="0" data-equation="|A|=1"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="A^{-1}=\begin{pmatrix}1&-2\\0&1\end{pmatrix}"><span></span><span></span></span>
2.4伴随矩阵<span class="equation-text" contenteditable="false" data-index="0" data-equation="A^*"><span></span><span></span></span>【近两年高频】
<span class="equation-text" contenteditable="false" data-index="0" data-equation="A^*"><span></span><span></span></span>构造规则：<span class="equation-text" contenteditable="false" data-index="0" data-equation="A_{ij}"><span></span><span></span></span>代数余子式<b>按转置摆放</b>
万能恒等式 <span class="equation-text" contenteditable="false" data-index="0" data-equation="\boldsymbol{AA^*=A^*A=|A|E}"><span></span><span></span></span>
衍生公式
<span class="equation-text" contenteditable="false" data-index="0" data-equation="(A^*)^{-1}=\dfrac{1}{|A|}A"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="|A^*|=|A|^{n-1}"><span></span><span></span></span>
📌2025‑10填空
二阶方阵<span class="equation-text" contenteditable="false" data-index="0" data-equation="|A|=3"><span></span><span></span></span>，<span class="equation-text" contenteditable="false" data-index="0" data-equation="AA^*=3E"><span></span><span></span></span>
2.5矩阵初等变换、初等矩阵
三类行变换：换行、数乘某行、倍加
初等矩阵左乘→行变换；右乘→列变换
求逆矩阵通用方法：<span class="equation-text" contenteditable="false" data-index="0" data-equation="(A\mid E) \xrightarrow{行变换}(E\mid A^{-1})"><span></span><span></span></span>
2.6矩阵的秩 r(A)
定义：最高阶非零子式阶数
性质
<span class="equation-text" contenteditable="false" data-index="0" data-equation="r(A^T)=r(A)"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="r(kA)=r(A),k\neq0"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="r(AB)\le \min(r(A),r(B))"><span></span><span></span></span>
n阶可逆矩阵 <span class="equation-text" contenteditable="false" data-index="0" data-equation="r(A)=n"><span></span><span></span></span>，满秩
✅求秩标准步骤
1）仅初等行变换化为行阶梯矩阵
2）非零行行数就是秩
⚠️禁止使用列变换求秩
