that det

199 days ago by William_Stein

def A(n): return matrix(ZZ,n,[j+1 if (i+1)%(j+1)==0 else 0 for i in range(n) for j in range(n)]) 
       
A(10) 
        
[ 1  0  0  0  0  0  0  0  0  0]
[ 1  2  0  0  0  0  0  0  0  0]
[ 1  0  3  0  0  0  0  0  0  0]
[ 1  2  0  4  0  0  0  0  0  0]
[ 1  0  0  0  5  0  0  0  0  0]
[ 1  2  3  0  0  6  0  0  0  0]
[ 1  0  0  0  0  0  7  0  0  0]
[ 1  2  0  4  0  0  0  8  0  0]
[ 1  0  3  0  0  0  0  0  9  0]
[ 1  2  0  0  5  0  0  0  0 10]
[ 1  0  0  0  0  0  0  0  0  0]
[ 1  2  0  0  0  0  0  0  0  0]
[ 1  0  3  0  0  0  0  0  0  0]
[ 1  2  0  4  0  0  0  0  0  0]
[ 1  0  0  0  5  0  0  0  0  0]
[ 1  2  3  0  0  6  0  0  0  0]
[ 1  0  0  0  0  0  7  0  0  0]
[ 1  2  0  4  0  0  0  8  0  0]
[ 1  0  3  0  0  0  0  0  9  0]
[ 1  2  0  0  5  0  0  0  0 10]
# sanity check A(10).det() == factorial(10) 
        
True
True
A(10)^(-1) 
        
[    1     0     0     0     0     0     0     0     0     0]
[ -1/2   1/2     0     0     0     0     0     0     0     0]
[ -1/3     0   1/3     0     0     0     0     0     0     0]
[    0  -1/4     0   1/4     0     0     0     0     0     0]
[ -1/5     0     0     0   1/5     0     0     0     0     0]
[  1/6  -1/6  -1/6     0     0   1/6     0     0     0     0]
[ -1/7     0     0     0     0     0   1/7     0     0     0]
[    0     0     0  -1/8     0     0     0   1/8     0     0]
[    0     0  -1/9     0     0     0     0     0   1/9     0]
[ 1/10 -1/10     0     0 -1/10     0     0     0     0  1/10]
[    1     0     0     0     0     0     0     0     0     0]
[ -1/2   1/2     0     0     0     0     0     0     0     0]
[ -1/3     0   1/3     0     0     0     0     0     0     0]
[    0  -1/4     0   1/4     0     0     0     0     0     0]
[ -1/5     0     0     0   1/5     0     0     0     0     0]
[  1/6  -1/6  -1/6     0     0   1/6     0     0     0     0]
[ -1/7     0     0     0     0     0   1/7     0     0     0]
[    0     0     0  -1/8     0     0     0   1/8     0     0]
[    0     0  -1/9     0     0     0     0     0   1/9     0]
[ 1/10 -1/10     0     0 -1/10     0     0     0     0  1/10]
show(A(10)^(-1)) 
        
\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrrrrrrrrr}
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
-\frac{1}{2} & \frac{1}{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
-\frac{1}{3} & 0 & \frac{1}{3} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & -\frac{1}{4} & 0 & \frac{1}{4} & 0 & 0 & 0 & 0 & 0 & 0 \\
-\frac{1}{5} & 0 & 0 & 0 & \frac{1}{5} & 0 & 0 & 0 & 0 & 0 \\
\frac{1}{6} & -\frac{1}{6} & -\frac{1}{6} & 0 & 0 & \frac{1}{6} & 0 & 0 & 0 & 0 \\
-\frac{1}{7} & 0 & 0 & 0 & 0 & 0 & \frac{1}{7} & 0 & 0 & 0 \\
0 & 0 & 0 & -\frac{1}{8} & 0 & 0 & 0 & \frac{1}{8} & 0 & 0 \\
0 & 0 & -\frac{1}{9} & 0 & 0 & 0 & 0 & 0 & \frac{1}{9} & 0 \\
\frac{1}{10} & -\frac{1}{10} & 0 & 0 & -\frac{1}{10} & 0 & 0 & 0 & 0 & \frac{1}{10}
\end{array}\right)
\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrrrrrrrrr}
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
-\frac{1}{2} & \frac{1}{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
-\frac{1}{3} & 0 & \frac{1}{3} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & -\frac{1}{4} & 0 & \frac{1}{4} & 0 & 0 & 0 & 0 & 0 & 0 \\
-\frac{1}{5} & 0 & 0 & 0 & \frac{1}{5} & 0 & 0 & 0 & 0 & 0 \\
\frac{1}{6} & -\frac{1}{6} & -\frac{1}{6} & 0 & 0 & \frac{1}{6} & 0 & 0 & 0 & 0 \\
-\frac{1}{7} & 0 & 0 & 0 & 0 & 0 & \frac{1}{7} & 0 & 0 & 0 \\
0 & 0 & 0 & -\frac{1}{8} & 0 & 0 & 0 & \frac{1}{8} & 0 & 0 \\
0 & 0 & -\frac{1}{9} & 0 & 0 & 0 & 0 & 0 & \frac{1}{9} & 0 \\
\frac{1}{10} & -\frac{1}{10} & 0 & 0 & -\frac{1}{10} & 0 & 0 & 0 & 0 & \frac{1}{10}
\end{array}\right)
html((A(20)^(-1)).str()) 
        
[    1     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0]
[ -1/2   1/2     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0]
[ -1/3     0   1/3     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0]
[    0  -1/4     0   1/4     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0]
[ -1/5     0     0     0   1/5     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0]
[  1/6  -1/6  -1/6     0     0   1/6     0     0     0     0     0     0     0     0     0     0     0     0     0     0]
[ -1/7     0     0     0     0     0   1/7     0     0     0     0     0     0     0     0     0     0     0     0     0]
[    0     0     0  -1/8     0     0     0   1/8     0     0     0     0     0     0     0     0     0     0     0     0]
[    0     0  -1/9     0     0     0     0     0   1/9     0     0     0     0     0     0     0     0     0     0     0]
[ 1/10 -1/10     0     0 -1/10     0     0     0     0  1/10     0     0     0     0     0     0     0     0     0     0]
[-1/11     0     0     0     0     0     0     0     0     0  1/11     0     0     0     0     0     0     0     0     0]
[    0  1/12     0 -1/12     0 -1/12     0     0     0     0     0  1/12     0     0     0     0     0     0     0     0]
[-1/13     0     0     0     0     0     0     0     0     0     0     0  1/13     0     0     0     0     0     0     0]
[ 1/14 -1/14     0     0     0     0 -1/14     0     0     0     0     0     0  1/14     0     0     0     0     0     0]
[ 1/15     0 -1/15     0 -1/15     0     0     0     0     0     0     0     0     0  1/15     0     0     0     0     0]
[    0     0     0     0     0     0     0 -1/16     0     0     0     0     0     0     0  1/16     0     0     0     0]
[-1/17     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0  1/17     0     0     0]
[    0     0  1/18     0     0 -1/18     0     0 -1/18     0     0     0     0     0     0     0     0  1/18     0     0]
[-1/19     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0  1/19     0]
[    0  1/20     0 -1/20     0     0     0     0     0 -1/20     0     0     0     0     0     0     0     0     0  1/20]
[    1     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0]
[ -1/2   1/2     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0]
[ -1/3     0   1/3     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0]
[    0  -1/4     0   1/4     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0]
[ -1/5     0     0     0   1/5     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0]
[  1/6  -1/6  -1/6     0     0   1/6     0     0     0     0     0     0     0     0     0     0     0     0     0     0]
[ -1/7     0     0     0     0     0   1/7     0     0     0     0     0     0     0     0     0     0     0     0     0]
[    0     0     0  -1/8     0     0     0   1/8     0     0     0     0     0     0     0     0     0     0     0     0]
[    0     0  -1/9     0     0     0     0     0   1/9     0     0     0     0     0     0     0     0     0     0     0]
[ 1/10 -1/10     0     0 -1/10     0     0     0     0  1/10     0     0     0     0     0     0     0     0     0     0]
[-1/11     0     0     0     0     0     0     0     0     0  1/11     0     0     0     0     0     0     0     0     0]
[    0  1/12     0 -1/12     0 -1/12     0     0     0     0     0  1/12     0     0     0     0     0     0     0     0]
[-1/13     0     0     0     0     0     0     0     0     0     0     0  1/13     0     0     0     0     0     0     0]
[ 1/14 -1/14     0     0     0     0 -1/14     0     0     0     0     0     0  1/14     0     0     0     0     0     0]
[ 1/15     0 -1/15     0 -1/15     0     0     0     0     0     0     0     0     0  1/15     0     0     0     0     0]
[    0     0     0     0     0     0     0 -1/16     0     0     0     0     0     0     0  1/16     0     0     0     0]
[-1/17     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0  1/17     0     0     0]
[    0     0  1/18     0     0 -1/18     0     0 -1/18     0     0     0     0     0     0     0     0  1/18     0     0]
[-1/19     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0  1/19     0]
[    0  1/20     0 -1/20     0     0     0     0     0 -1/20     0     0     0     0     0     0     0     0     0  1/20]
html((A(30)^(-1)).str()) 
        
[    1     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0]
[ -1/2   1/2     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0]
[ -1/3     0   1/3     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0]
[    0  -1/4     0   1/4     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0]
[ -1/5     0     0     0   1/5     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0]
[  1/6  -1/6  -1/6     0     0   1/6     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0]
[ -1/7     0     0     0     0     0   1/7     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0]
[    0     0     0  -1/8     0     0     0   1/8     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0]
[    0     0  -1/9     0     0     0     0     0   1/9     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0]
[ 1/10 -1/10     0     0 -1/10     0     0     0     0  1/10     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0]
[-1/11     0     0     0     0     0     0     0     0     0  1/11     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0]
[    0  1/12     0 -1/12     0 -1/12     0     0     0     0     0  1/12     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0]
[-1/13     0     0     0     0     0     0     0     0     0     0     0  1/13     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0]
[ 1/14 -1/14     0     0     0     0 -1/14     0     0     0     0     0     0  1/14     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0]
[ 1/15     0 -1/15     0 -1/15     0     0     0     0     0     0     0     0     0  1/15     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0]
[    0     0     0     0     0     0     0 -1/16     0     0     0     0     0     0     0  1/16     0     0     0     0     0     0     0     0     0     0     0     0     0     0]
[-1/17     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0  1/17     0     0     0     0     0     0     0     0     0     0     0     0     0]
[    0     0  1/18     0     0 -1/18     0     0 -1/18     0     0     0     0     0     0     0     0  1/18     0     0     0     0     0     0     0     0     0     0     0     0]
[-1/19     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0  1/19     0     0     0     0     0     0     0     0     0     0     0]
[    0  1/20     0 -1/20     0     0     0     0     0 -1/20     0     0     0     0     0     0     0     0     0  1/20     0     0     0     0     0     0     0     0     0     0]
[ 1/21     0 -1/21     0     0     0 -1/21     0     0     0     0     0     0     0     0     0     0     0     0     0  1/21     0     0     0     0     0     0     0     0     0]
[ 1/22 -1/22     0     0     0     0     0     0     0     0 -1/22     0     0     0     0     0     0     0     0     0     0  1/22     0     0     0     0     0     0     0     0]
[-1/23     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0  1/23     0     0     0     0     0     0     0]
[    0     0     0  1/24     0     0     0 -1/24     0     0     0 -1/24     0     0     0     0     0     0     0     0     0     0     0  1/24     0     0     0     0     0     0]
[    0     0     0     0 -1/25     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0  1/25     0     0     0     0     0]
[ 1/26 -1/26     0     0     0     0     0     0     0     0     0     0 -1/26     0     0     0     0     0     0     0     0     0     0     0     0  1/26     0     0     0     0]
[    0     0     0     0     0     0     0     0 -1/27     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0  1/27     0     0     0]
[    0  1/28     0 -1/28     0     0     0     0     0     0     0     0     0 -1/28     0     0     0     0     0     0     0     0     0     0     0     0     0  1/28     0     0]
[-1/29     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0  1/29     0]
[-1/30  1/30  1/30     0  1/30 -1/30     0     0     0 -1/30     0     0     0     0 -1/30     0     0     0     0     0     0     0     0     0     0     0     0     0     0  1/30]
[    1     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0]
[ -1/2   1/2     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0]
[ -1/3     0   1/3     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0]
[    0  -1/4     0   1/4     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0]
[ -1/5     0     0     0   1/5     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0]
[  1/6  -1/6  -1/6     0     0   1/6     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0]
[ -1/7     0     0     0     0     0   1/7     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0]
[    0     0     0  -1/8     0     0     0   1/8     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0]
[    0     0  -1/9     0     0     0     0     0   1/9     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0]
[ 1/10 -1/10     0     0 -1/10     0     0     0     0  1/10     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0]
[-1/11     0     0     0     0     0     0     0     0     0  1/11     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0]
[    0  1/12     0 -1/12     0 -1/12     0     0     0     0     0  1/12     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0]
[-1/13     0     0     0     0     0     0     0     0     0     0     0  1/13     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0]
[ 1/14 -1/14     0     0     0     0 -1/14     0     0     0     0     0     0  1/14     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0]
[ 1/15     0 -1/15     0 -1/15     0     0     0     0     0     0     0     0     0  1/15     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0]
[    0     0     0     0     0     0     0 -1/16     0     0     0     0     0     0     0  1/16     0     0     0     0     0     0     0     0     0     0     0     0     0     0]
[-1/17     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0  1/17     0     0     0     0     0     0     0     0     0     0     0     0     0]
[    0     0  1/18     0     0 -1/18     0     0 -1/18     0     0     0     0     0     0     0     0  1/18     0     0     0     0     0     0     0     0     0     0     0     0]
[-1/19     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0  1/19     0     0     0     0     0     0     0     0     0     0     0]
[    0  1/20     0 -1/20     0     0     0     0     0 -1/20     0     0     0     0     0     0     0     0     0  1/20     0     0     0     0     0     0     0     0     0     0]
[ 1/21     0 -1/21     0     0     0 -1/21     0     0     0     0     0     0     0     0     0     0     0     0     0  1/21     0     0     0     0     0     0     0     0     0]
[ 1/22 -1/22     0     0     0     0     0     0     0     0 -1/22     0     0     0     0     0     0     0     0     0     0  1/22     0     0     0     0     0     0     0     0]
[-1/23     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0  1/23     0     0     0     0     0     0     0]
[    0     0     0  1/24     0     0     0 -1/24     0     0     0 -1/24     0     0     0     0     0     0     0     0     0     0     0  1/24     0     0     0     0     0     0]
[    0     0     0     0 -1/25     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0  1/25     0     0     0     0     0]
[ 1/26 -1/26     0     0     0     0     0     0     0     0     0     0 -1/26     0     0     0     0     0     0     0     0     0     0     0     0  1/26     0     0     0     0]
[    0     0     0     0     0     0     0     0 -1/27     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0  1/27     0     0     0]
[    0  1/28     0 -1/28     0     0     0     0     0     0     0     0     0 -1/28     0     0     0     0     0     0     0     0     0     0     0     0     0  1/28     0     0]
[-1/29     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0  1/29     0]
[-1/30  1/30  1/30     0  1/30 -1/30     0     0     0 -1/30     0     0     0     0 -1/30     0     0     0     0     0     0     0     0     0     0     0     0     0     0  1/30]
html((A(50)^(-1)).str()) 
        
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[    0     0     0  1/24     0     0     0 -1/24     0     0     0 -1/24     0     0     0     0     0     0     0     0     0     0     0  1/24     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0]
[    0     0     0     0 -1/25     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0  1/25     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0]
[ 1/26 -1/26     0     0     0     0     0     0     0     0     0     0 -1/26     0     0     0     0     0     0     0     0     0     0     0     0  1/26     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0]
[    0     0     0     0     0     0     0     0 -1/27     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0  1/27     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0]
[    0  1/28     0 -1/28     0     0     0     0     0     0     0     0     0 -1/28     0     0     0     0     0     0     0     0     0     0     0     0     0  1/28     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0]
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[-1/30  1/30  1/30     0  1/30 -1/30     0     0     0 -1/30     0     0     0     0 -1/30     0     0     0     0     0     0     0     0     0     0     0     0     0     0  1/30     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0]
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[    0     0     0     0     0     0     0     0     0     0     0     0     0     0     0 -1/32     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0  1/32     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0]
[ 1/33     0 -1/33     0     0     0     0     0     0     0 -1/33     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0  1/33     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0]
[ 1/34 -1/34     0     0     0     0     0     0     0     0     0     0     0     0     0     0 -1/34     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0  1/34     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0]
[ 1/35     0     0     0 -1/35     0 -1/35     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0  1/35     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0]
[    0     0     0     0     0  1/36     0     0     0     0     0 -1/36     0     0     0     0     0 -1/36     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0  1/36     0     0     0     0     0     0     0     0     0     0     0     0     0     0]
[-1/37     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0  1/37     0     0     0     0     0     0     0     0     0     0     0     0     0]
[ 1/38 -1/38     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0 -1/38     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0  1/38     0     0     0     0     0     0     0     0     0     0     0     0]
[ 1/39     0 -1/39     0     0     0     0     0     0     0     0     0 -1/39     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0  1/39     0     0     0     0     0     0     0     0     0     0     0]
[    0     0     0  1/40     0     0     0 -1/40     0     0     0     0     0     0     0     0     0     0     0 -1/40     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0  1/40     0     0     0     0     0     0     0     0     0     0]
[-1/41     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0  1/41     0     0     0     0     0     0     0     0     0]
[-1/42  1/42  1/42     0     0 -1/42  1/42     0     0     0     0     0     0 -1/42     0     0     0     0     0     0 -1/42     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0  1/42     0     0     0     0     0     0     0     0]
[-1/43     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0  1/43     0     0     0     0     0     0     0]
[    0  1/44     0 -1/44     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0 -1/44     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0  1/44     0     0     0     0     0     0]
[    0     0  1/45     0     0     0     0     0 -1/45     0     0     0     0     0 -1/45     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0  1/45     0     0     0     0     0]
[ 1/46 -1/46     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0 -1/46     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0  1/46     0     0     0     0]
[-1/47     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0  1/47     0     0     0]
[    0     0     0     0     0     0     0  1/48     0     0     0     0     0     0     0 -1/48     0     0     0     0     0     0     0 -1/48     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0  1/48     0     0]
[    0     0     0     0     0     0 -1/49     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0  1/49     0]
[    0     0     0     0  1/50     0     0     0     0 -1/50     0     0     0     0     0     0     0     0     0     0     0     0     0     0 -1/50     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0  1/50]

How fast to invert in Sage?
time B = A(100) 
        
Time: CPU 0.01 s, Wall: 0.01 s
Time: CPU 0.01 s, Wall: 0.01 s
time C = B^(-1) 
        
Time: CPU 0.31 s, Wall: 0.33 s
Time: CPU 0.31 s, Wall: 0.33 s
time B = A(200) 
        
Time: CPU 0.04 s, Wall: 0.04 s
Time: CPU 0.04 s, Wall: 0.04 s
time C = B^(-1) 
        
Time: CPU 1.39 s, Wall: 1.39 s
Time: CPU 1.39 s, Wall: 1.39 s