# Lecture 03 Multiplication and Inverse Matrices

Today's lecture includes:
1. Matrix Multiplication (4 ways)
2. Inverse of $A, AB, A^{T}$
3. Gauss-Jordan / Find $A^{-1}$

## Matrix Multiplication
### 1.1 First method

We first take a look on this matrix:
\begin{align}
\begin{matrix}-&-&-\\-&-&-&\end{matrix} × \begin{matrix}-&-\\-&-\\-&-\end{matrix} = \begin{matrix}-&-\\-&-\end{matrix}
\end{align}
* 行列内积：有$m\times n$矩阵$A$和$n\times p$矩阵$B$（$A$的总行数必须与$B$的总列数相等），两矩阵相乘有$AB=C$，$C$是一个$m\times p$矩阵，对于$C$矩阵中的第$i$行第$j$列元素$c_{ij}$，有：

    $$c_{ij}=row_i\cdot column_j=\sum_{k=i}^na_{ik}b_{kj}$$

    其中$a_{ik}$是$A$矩阵的第$i$行第$k$列元素，$b_{kj}$是$B$矩阵的第$k$行第$j$列元素。

    可以看出$c_{ij}$其实是$A$矩阵第$i$行点乘$B$矩阵第$j$列 $\begin{bmatrix}&\vdots&\\&row_i&\\&\vdots&\end{bmatrix}\begin{bmatrix}&&\\\cdots&column_j&\cdots\\&&\end{bmatrix}=\begin{bmatrix}&\vdots&\\\cdots&c_{ij}&\cdots\\&\vdots&\end{bmatrix}$


```python

```

10bb9ffb467f26b725c5fc09e70327c53f400fe6
ipynb
Jupyter Notebook
Lecture 03 Multiplication and Inverse Matrices.ipynb
XingxinHE/Linear_Algebra
7d6b78699f8653ece60e07765fd485dd36b26194
MIT
2021-04-24T17:23:50.000Z
2021-11-27T11:00:04.000Z
Lecture 03 Multiplication and Inverse Matrices.ipynb
XingxinHE/Linear_Algebra
7d6b78699f8653ece60e07765fd485dd36b26194
MIT
Lecture 03 Multiplication and Inverse Matrices.ipynb
XingxinHE/Linear_Algebra
7d6b78699f8653ece60e07765fd485dd36b26194
MIT
Qwen/Qwen-72B
1. YES
2. YES