From 201f38a1a1300a945da3035f774d1abc33062bb2 Mon Sep 17 00:00:00 2001
From: unknown <18951088369@163.com>
Date: Tue, 30 Dec 2025 23:15:00 +0800
Subject: [PATCH] vault backup: 2025-12-30 23:15:00

---
 编写小组/试卷/1231线性代数考试卷.md | 11 ++++++++++-
 1 file changed, 10 insertions(+), 1 deletion(-)

diff --git a/编写小组/试卷/1231线性代数考试卷.md b/编写小组/试卷/1231线性代数考试卷.md
index 1598da1..9983091 100644
--- a/编写小组/试卷/1231线性代数考试卷.md
+++ b/编写小组/试卷/1231线性代数考试卷.md
@@ -53,7 +53,16 @@ tags:
 
 10. 设矩阵$A = \begin{bmatrix}1 & a_1 & a_1^2 & a_1^3 \\1 & a_2 & a_2^2 & a_2^3 \\1 & a_3 & a_3^2 & a_3^3 \\1 & a_4 & a_4^2 & a_4^3\end{bmatrix},x = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{bmatrix},b = \begin{bmatrix} 1 \\ 1 \\ 1 \\ 1 \end{bmatrix},$
     其中常数 $a_1, a_2, a_3, a_4$ 互不相等，则线性方程组 $Ax = b$ 的解为$\underline{\qquad\qquad\qquad\qquad}.$
-11. $A^{k} = 0, k=\underline{\qquad\qquad\qquad\qquad}.$
+11. 矩阵$$A=\begin{bmatrix}
+0 & 0 & \cdots & 1 & 1 & \cdots & 1 & 1 \\
+0 & 0 & \cdots & 0 & 1 & \cdots & 1 & 1 \\
+\vdots & \vdots & \ddots & \vdots & \vdots & \ddots & \vdots & \vdots \\
+0 & 0 & \cdots & 0 & 0 & \cdots & 1 & 1 \\
+0 & 0 & \cdots & 0 & 0 & \cdots & 0 & 1 \\
+\vdots & \vdots & \ddots & \vdots & \vdots & \ddots & \vdots & \vdots \\
+0 & 0 & \cdots & 0 & 0 & \cdots & 0 & 0 \\
+0 & 0 & \cdots & 0 & 0 & \cdots & 0 & 0
+\end{bmatrix}_{n \times n}$$其中第一行有$m$个$0$.若$A^k=0$,则$k$的最小值为____.$A^{k} = 0, k=\underline{\qquad\qquad\qquad\qquad}.$
 12. $\underline{\qquad\qquad\qquad\qquad}$
 
 ## 三、解答题，共五道，共64分