diff --git a/linear-algebra/exercise/1-determinant/determinant.pdf b/linear-algebra/exercise/1-determinant/determinant.pdf
index b70ff7a..11f64df 100644
Binary files a/linear-algebra/exercise/1-determinant/determinant.pdf and b/linear-algebra/exercise/1-determinant/determinant.pdf differ
diff --git a/linear-algebra/exercise/1-determinant/determinant.tex b/linear-algebra/exercise/1-determinant/determinant.tex
index 5cfa286..04fc947 100644
--- a/linear-algebra/exercise/1-determinant/determinant.tex
+++ b/linear-algebra/exercise/1-determinant/determinant.tex
@@ -86,7 +86,7 @@ $\therefore\,-a_{11}a_{23}a_{32}a_{44}+a_{11}a_{23}a_{34}a_{42}$。
 
 \subsection{基本行列式与计算}
 
-\subsubsection{三角行列式}
+\subsubsection{主对角线行列式}
 
 $\left|\begin{array}{cccc} 
     a_{11} & a_{12} & \cdots & a_{1n} \\
@@ -107,7 +107,7 @@ $\left|\begin{array}{cccc}
      & & & a_{nn}
 \end{array}\right|=a_{11}\cdots a_{nn}$
 
-\subsubsection{反三角行列式}
+\subsubsection{副对角线行列式}
 
 $\left|\begin{array}{cccc} 
     & & & a_{1n} \\
@@ -135,10 +135,12 @@ $\left|\begin{array}{cccc}
     a_1 & a_2  & \cdots & a_n \\
     \cdots & \cdots & \vdots & \cdots \\
     a_1^{n-1} & a_2^{n-1} & \cdots & a_n^{n-1} \\
-\end{array}\right|=\sum\limits_{1\leqslant j<i\leqslant n}(a_i-a_j)$
+\end{array}\right|=\prod\limits_{1\leqslant j<i\leqslant n}(a_i-a_j)$
 
 \subsubsection{分块行列式}
 
+也称为拉普拉斯展开式，设$A$为$m$阶矩阵，$B$为$n$阶矩阵：
+
 $\left|\begin{array}{cc}
     A & O \\
     O & B
@@ -152,6 +154,19 @@ $\left|\begin{array}{cc}
     * & B
 \end{array}\right|=\vert A\vert\cdot\vert B\vert$。
 
+$\left|\begin{array}{cc}
+    O & A \\
+    B & O
+\end{array}\right|=
+\left|\begin{array}{cc}
+    * & A \\
+    B & O
+\end{array}\right|=
+\left|\begin{array}{cc}
+    O & A \\
+    B & *
+\end{array}\right|=(-1)^{mn}\vert A\vert\cdot\vert B\vert$。
+
 \subsubsection{基本行列式计算}
 
 基本的计算方式是对角线法则计算与行列式展开两种方法。若符合基本特殊行列式的可以按照公式。
@@ -163,6 +178,25 @@ $\left|\begin{array}{cc}
     \item 通过行列式的对换从上往下让行列式变成上三角行列式，对角线相乘就得到结果。
 \end{itemize}
 
+\textbf{例题：}计算$\left|\begin{array}{cccccc} 
+    a & b & 0 & \cdots & 0 & 0 \\
+    0 & a & b & \cdots & 0 & 0 \\
+    0 & 0 & a & \cdots & 0 & 0 \\
+    \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\
+    0 & 0 & 0 & \cdots & a & b \\
+    b & 0 & 0 & \cdots & 0 & a
+\end{array}\right|$。
+
+直接按第一列展开：
+
+$=a(-1)^{1+1}\left|\begin{array}{ccccc} 
+    a & b & \cdots & 0 & 0 \\
+    0 & a & \cdots & 0 & 0 \\
+    \vdots & \vdots & \ddots & \vdots & \vdots \\
+    0 & 0 & \cdots & a & b \\
+    0 & 0 & \cdots & 0 & a
+\end{array}\right|$
+
 \subsection{提取公因式}
 
 可以提取某一行或某一列的公因式。
diff --git a/linear-algebra/knowledge/1-determinant/determinant.pdf b/linear-algebra/knowledge/1-determinant/determinant.pdf
index bee1d00..11fefa5 100644
Binary files a/linear-algebra/knowledge/1-determinant/determinant.pdf and b/linear-algebra/knowledge/1-determinant/determinant.pdf differ
diff --git a/linear-algebra/knowledge/1-determinant/determinant.tex b/linear-algebra/knowledge/1-determinant/determinant.tex
index 7bdc41e..c9cb538 100644
--- a/linear-algebra/knowledge/1-determinant/determinant.tex
+++ b/linear-algebra/knowledge/1-determinant/determinant.tex
@@ -132,7 +132,7 @@ $\alpha_1=[a_{11},a_{12},\cdots,a_{1n}]$，$\alpha_2=[a_{21},a_{22},\cdots,a_{2n
 
 \subsection{特殊行列式}
 
-\subsubsection{三角行列式}
+\subsubsection{主对角线行列式}
 
 $\left|\begin{array}{cccc} 
     a_{11} & a_{12} & \cdots & a_{1n} \\
@@ -155,13 +155,11 @@ $\left|\begin{array}{cccc}
 
 上三角行列式：包括主对角线的右上部分元素不全为0，左下部分元素全为0。
 
-
 下三角行列式：包括主对角线的左下部分元素不全为0，右上部分元素全为0。
 
-
 对角行列式：省略号处的元素不全为0，其他主对角线外的元素全为0。
 
-\subsubsection{反三角行列式}
+\subsubsection{副对角线行列式}
 
 $\left|\begin{array}{cccc} 
      & & & a_{1n} \\
@@ -195,7 +193,7 @@ $\left|\begin{array}{cccc}
         a_1^{n-1} & a_2^{n-1} & \cdots & a_n^{n-1} \\
     \end{array}\right|$
 
-    范德蒙德行列式：元素连乘，结果为$\sum\limits_{1\leqslant j<i\leqslant n}(a_i-a_j)$。
+    范德蒙德行列式：元素连乘，结果为$\prod\limits_{1\leqslant j<i\leqslant n}(a_i-a_j)$。
     若一个四阶范德蒙德行列式的结果为$(a_4-a_1)(a_4-a_2)(a_4-a_3)(a_3-a_1)(a_3-a_2)(a_2-a_1)$。
 
 \end{multicols}