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Let $\omega =e^{i\pi /7}$, a primitive fourteenth root of unity. The geometric series gives ${\sum }_{k=1}^{14}{\omega }^k=0$. Taking the real part and adding the remaining fifteenth term therefore yieldsThe upper summation limit matters: the first fourteen terms complete one full cycle, and the fifteenth supplies one additional term.

Rearrange as $x^{\prime \prime }-4x^{\prime }+3x=10t$. The characteristic polynomial factors as $(r-1)(r-3)$, giving the homogeneous solutions $e^t,e^{3t}$. A particular polynomial solution $At+B$ obeys $3A=10$ and $3B-4A=0$. HenceThis is the full two-constant solution of the second-order linear differential equation.

The elementary row operation $R_2\leftarrow R_2+3R_1$ givesThe nonzero rows here have distinct pivots. Therefore the matrix rank isThere is always at least one nonzero row, so rank zero is impossible.

Write $z=x+iy$ with real $x,y$. The complex hyperbolic sine is $\sinh z=\sinh x\cos y+i\cosh x\sin y$. Its zero real part requires $\sinh x\cos y=0$. If $\cos y=0$, the magnitude of the imaginary part would be $\cosh x\ge 1$, which cannot equal $1/2$. Therefore $x=0$ and $\sin y=1/2$. All solutions areOn the Argand diagram these are two interlaced arithmetic sequences on the imaginary axis; both repeat with imaginary period $2\pi i$.

The determinant is $a^2-i(-i)=a^2-1$. The adjugate matrix formula givesWhen $a=1$ or $a=-1$ we have a singular matrix, so no inverse matrix exists. This criterion holds for real or complex $a$.

Multiply numerator and denominator by the denominator's complex conjugate:Thus the real part is $\boxed{7/41}$ and the imaginary part is $\boxed{22/41}$.