Determinant of a matrix having diagonal and subdiagonal entries zero

Here is a principal solution (some details left for you).

Let $A$ be an $n\times n$ tridiagonal matrix such that all its entries consisting of zeros except for those on (i) the main and subdiagonals are $-1$; (ii) superdiagonals are $-2$.

Let $u$ be the column vector all entries are $1$ so that $uu^T$ is an $n\times n$ matrix of all $1$'s. This way, your matrix becomes $A+uu^T$. Now, apply the Matrix Determinant Lemma to obtain $$\det(A+uu^T)=(1+u^TA^{-1}u)\cdot\det(A).$$ The good news is: determinants and inverses of tridiagonal matrices are calculable (see this on Wiki). For example, in the present case $$f_n:=\det(A)=(-1)^n\sum_{k=0}^{\lfloor\frac{n}2\rfloor}(-1)^k\binom{n-k}k2^k.$$ Also, observe that the quantity $u^TA^{-1}u$ is the sum of all entries of the inverse matrix $A^{-1}$.

UPDATE. Since you still ask about the inverse and etc, I've computed your determinant \begin{align} \det(A+uu^T)&=f_n+\sum_{i\leq j}^{1,n}2^{j-i}f_{i-1}f_{n-j}+\sum_{j<i}^{1,n}f_{j-1}f_{n-i} \\ &=f_n+\sum_{i=1}^nf_{i-1}f_{n-i}+\sum_{j<i}^{1,n}(1+2^{i-j})f_{j-1}f_{n-i}. \end{align}

I will try to give a different explicit solution. From the problem formulation of answer 1, $$\det(A+uu^T)=(1+u^TA^{-1}u).\det(A).$$ From here, we need to solve some recursive expressions, in order to calculate the determinant and inverse of $A$. I solve these recursive expressions using roots of their characteristic equations. For the determinant of $A$, recursive expression is $$f_n=-f_{n-1}-2f_{n-2},\ \ \ \ \ f_0=1,\ f_{-1}=0.$$ Roots of the resulting characteristic equation $x^2+x+2=0,$ are $$ s_1=\frac{-1}{2}+ \frac{\iota \sqrt{7}}{2},\ \ s_2=\frac{-1}{2}- \frac{\iota \sqrt{7}}{2}.$$ Hence, $$\det(A)=f_n=r_1.s_1^n+r_2.s_2^n,$$ where, using initial conditions $$r_1=\frac{1}{2} +\frac{\iota }{2\sqrt{7}},\ \ r_2=\frac{1}{2} -\frac{\iota }{2\sqrt{7}}.$$ Now, to calculate $A^{-1}$ we need to solve following recursive expressions $$g_i=-g_{i-1}-2g_{i-1}, \ \text{for} \ \ i=2, 3\dots,n,\ \ g_0=1,\ g_1=-1$$ $$ h_i=-h_{i+1}-2h_{i+2}, \ \text{for} \ \ i=n-1,\dots,1, \ \ h_{n+1}=1,\ h_{n}=-1.$$ Similar to $f_n$, solving these recursive expressions we get $$g_i=r_{1}.s_1^i+r_{2}.s_2^i.$$ And, $$h_i=r_{h1}.s_1^{n-i}+r_{h2}.s_2^{n-i},$$ where, $$r_{h1}=\frac{-1}{2}-\frac{3}{2\iota\sqrt(7)},\ \ \ r_{h2}=\frac{-1}{2}+\frac{3}{2\iota\sqrt(7)}.$$ Entries of $A^{-1}$ are clearly given by $g_i, h_i$ (here).

$$A^{-1}_{ij}=\begin{cases}\frac{2^{j-i}g_{i-1}h_{j+1}}{g_n} & \mbox{if $i\le j$} \\ \frac{g_{j-1}h_{i+1}}{g_n} &\mbox{if $j< i$} \end{cases}.$$