设 4 阶矩阵

$$
B=\left(\begin{array}{cccc}
1 & -1 & 0 & 0 \\
0 & 1 & -1 & 0 \\
0 & 0 & 1 & -1 \\
0 & 0 & 0 & 0
\end{array}\right), C=\left(\begin{array}{llll}
2 & 1 & 3 & 4 \\
0 & 2 & 1 & 3 \\
0 & 0 & 2 & 1 \\
0 & 0 & 0 & 2
\end{array}\right)
$$


且矩阵 $A$ 满足关系式 $A\left(I-C^{-1} B\right)^{ T } C^{ T }=I$ ，求矩阵 $A$ ．