分块初等变换与降阶公式#
这一节介绍分块初等变换在求行列式值和求逆的应用
65. 设A是n阶可逆阵，$ α,β$是n维列向量，b是常数，现有分块矩阵#
$$
Q=\left(\begin{array}{cccc}
A &α\\
β’ &b\\
\end{array}\right)
$$
证明：矩阵Q是可逆阵的充要条件是$b≠β’A^{-1}α$
66. (行列式的降阶公式)#
设A是m阶矩阵，D是n阶矩阵，B是$m×n$阶矩阵，C是$n×m$阶矩阵，证明：
(1) 若A可逆，则
$$
\left|\begin{array}{cccc}
A &B\\
C &D\\
\end{array}\right|
=|A||D-CA^{-1}B|
$$
(2) 若D可逆，则
$$
\left|\begin{array}{cccc}
A &B\\
C &D\\
\end{array}\right|
=|D||A-BD^{-1}C|
$$
(3) 若A,D都可逆，则
$$
|D||A-BD^{-1}C|=|A||D-CA^{-1}B|
$$
67. 求下列矩阵的行列式的值#
$$
A=\left(\begin{array}{cccc}
a_{1}^{2} &a_{1}a_{2}+1 &\cdots &a_{1}a_{n}+1\\
a_{2}^{1}+1 &a_{2}a_{2} &\cdots &a_{2}a_{n}+1\\
\vdots &\vdots & &\vdots\\
a_{n}^{2}+1 &a_{n}a_{2}+1 &\cdots &a_{n}a_{2}\\
\end{array}\right)
$$
68. 求下列矩阵的行列式的值#
$$
A=\left(\begin{array}{cccc}
0 &2 &3 &\cdots &n\\
1 &0 &3 &\cdots &n\\
1 &2 &0 &\dots&n\\
\vdots &\vdots &\vdots & &\vdots\\
1 &2 &3 &\cdots &0\\
\end{array}\right)
$$
1.33 求下列矩阵的值，其中$a_{i}≠0(1 \le i \le n)$;#
$$
A=\left(\begin{array}{cccc}
0 &a_{1}+a_{2} &\cdots &a_{1}+a_{n}\\
a_{2}+a_{1} &0 &\cdots &a_{2}+a_{n}\\
\vdots &\vdots & &\vdots\\
a_{n}+a_{1} &a_{n}+a_{2} &\cdots &0\\
\end{array}\right)
$$
69. 设A,B是n阶矩阵，求证：#
$$
\left|\begin{array}{cccc}
A &B\\
B &A\\
\end{array}\right|
=|A+B||A-B|
$$
1.36. 计算#
$$
|A|=\left|\begin{array}{cccc}
x &y &z &w\\
y &x &w &z\\
z &w &x &y\\
w &z &y &x\\
\end{array}\right|
$$
70. 设A,B,C,D都是n阶矩阵，求证：#
$$
|M|=\left|\begin{array}{cccc}
A &B &C &D\\
B &A &D &C\\
C &D &A &B\\
D &C &A &B\\
\end{array}\right|
=|A+B+C+D||A+B-C-D||A-B+C-D||A-B-C+D|
$$
71. 设A,B是n阶复矩阵，求证：#
$$
\left|\begin{array}{cccc}
A &-B\\
B &A \\
\end{array}\right|
=|A+iB||A-iB|
$$
72. 设A,B是n阶矩阵且AB=BA，求证：#
$$
\left|\begin{array}{cccc}
A &-B\\
B &A \\
\end{array}\right|
=|A^{2}+B^{2}|
$$
73. 设A,B是n阶实矩阵，求证：#
$$
\left|\begin{array}{cccc}
A &-B\\
B &A \\
\end{array}\right|
\ge 0
$$
2.55 求下列矩阵的行列式的值#
$$
A=\left(\begin{array}{cccc}
x &-y &-z &-w\\
y &x &-w &z\\
z &w &x &-y\\
w &-z &y &x\\
\end{array}\right)
$$
74. 已知A和D是可逆阵，求下列分块矩阵的逆阵#
$$
\left(\begin{array}{cccc}
A &B\\
O &D\\
\end{array}\right)
$$