If $$A = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}$$, then the determinant of $$A + A^2 + A^3 + \cdots + A^{13}$$ is

By induction, $$A^n = \begin{bmatrix} 1 & n \\ 0 & 1 \end{bmatrix}$$.

$$\sum_{n=1}^{13} A^n = \begin{bmatrix} 13 & 1+2+\cdots+13 \\ 0 & 13 \end{bmatrix} = \begin{bmatrix} 13 & 91 \\ 0 & 13 \end{bmatrix}$$.

Determinant $$= 13 \cdot 13 - 91 \cdot 0 = \mathbf{169}$$.

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