messages listlengths 1 1 | ground_truth stringclasses 3
values | dataset stringclasses 1
value |
|---|---|---|
[
{
"content": "\\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).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | n | MATH |
[
{
"content": "\\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).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | n | MATH |
[
{
"content": "\\text{Let } C_n \\text{ be } n \\times n, \\ c_{ij} = i \\times j. \\text{ Find } \\mathrm{rank}(C_n).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 1 | MATH |
[
{
"content": "\\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).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | n | MATH |
[
{
"content": "\\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).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 2 | MATH |
[
{
"content": "\\text{Let } F_n \\text{ be } n \\times n, \\ f_{ij} = 2^{i + j}. \\text{ Find } \\mathrm{rank}(F_n).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 1 | MATH |
[
{
"content": "\\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).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 2 | MATH |
[
{
"content": "\\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).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "us... | 1 | MATH |
[
{
"content": "\\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).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | n | MATH |
[
{
"content": "\\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).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | n | MATH |
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