+64 Find the ordered quintuplet that satisfies the system of equations
\(a+2b+3c+4d+5e=47 \)
\(2a+3b+4c+5d+e=15\)
\(3a+4b+5c+d+2e=3\)
\(4a+5b+c+2d+3e=26\)
\(5a+b+2c+3d+4e=20\)
+1588 We can write the equation as
\[
\begin{pmatrix}
1 & 2 & 3 & 4 & 5 \\
2 & 3 & 4 & 5 & 1 \\
3 & 4 & 5 & 1 & 2 \\
4 & 5 & 1 & 2 & 3 \\
5 & 1 & 2 & 3 & 4
\end{pmatrix}
\begin{pmatrix}
a \\ b \\ c \\ d \\ e
\end{pmatrix}
=
\begin{pmatrix}
47 \\ 15 \\ 3 \\ 26 \\ 29
\end{pmatrix}
.
\]
Inverting the matrix, we get the solution (a,b,c,d,e) = (-5, 4, 14, 3, -2).
