[User]
\text{Let } A_n \text{ be an } n \times n \text{ matrix where } a_{ij} = 1 \text{ if } i \leq j, \text{ else } 0. \text{ Find } \mathrm{rank}(A_n).

Present the answer in LaTex format: \boxed{Your answer}

ground_truth: n

dataset: MATH


[User]
\text{Let } B_n \text{ be } n \times n, \ b_{ij} = 1 \text{ if } i + j \leq n + 1, \text{ else } 0. \text{ Find } \mathrm{rank}(B_n).

Present the answer in LaTex format: \boxed{Your answer}


[User]
\text{Let } C_n \text{ be } n \times n, \ c_{ij} = i \times j. \text{ Find } \mathrm{rank}(C_n).

Present the answer in LaTex format: \boxed{Your answer}


[User]
\text{Let } D_n \text{ be } n \times n, \ d_{ij} = \begin{cases} 1 & \text{if } i = j \\ 2 & \text{if } |i - j| = 1 \\ 0 & \text{otherwise} \end{cases}. \text{ Find } \mathrm{rank}(D_n).

Present the answer in LaTex format: \boxed{Your answer}


[User]
\text{Let } E_n \text{ be } n \times n, \ e_{ij} = \begin{cases} 1 & \text{if } i + j \text{ is even} \\ 0 & \text{if } i + j \text{ is odd} \end{cases}. \text{ Find } \mathrm{rank}(E_n).

Present the answer in LaTex format: \boxed{Your answer}


[User]
\text{Let } F_n \text{ be } n \times n, \ f_{ij} = 2^{i + j}. \text{ Find } \mathrm{rank}(F_n).

Present the answer in LaTex format: \boxed{Your answer}


[User]
\text{Let } G_n \text{ be } n \times n, \ g_{ij} = \begin{cases} 1 & \text{if } i = 1 \text{ or } j = 1 \\ 0 & \text{otherwise} \end{cases}. \text{ Find } \mathrm{rank}(G_n).

Present the answer in LaTex format: \boxed{Your answer}


[User]
\text{Let } H_n \text{ be } n \times n, \ h_{ij} = \begin{cases} 1 & \text{if } i \leq n/2 \text{ and } j \leq n/2 \\ 0 & \text{otherwise} \end{cases} \ (\text{n even}). \text{ Find } \mathrm{rank}(H_n).

Present the answer in LaTex format: \boxed{Your answer}


[User]
\text{Let } I_n \text{ be } n \times n, \ i_{ij} = \begin{cases} i & \text{if } i = j \\ 0 & \text{otherwise} \end{cases}. \text{ Find } \mathrm{rank}(I_n).

Present the answer in LaTex format: \boxed{Your answer}


[User]
\text{Let } J_n \text{ be } n \times n, \ j_{ij} = \begin{cases} 1 & \text{if } i + j = n + 1 \\ 0 & \text{otherwise} \end{cases}. \text{ Find } \mathrm{rank}(J_n).

Present the answer in LaTex format: \boxed{Your answer}