Matrix Pow Sum

AtCoder

IDarc190_d

Time2000ms

Memory256MB

Difficulty—

You are given a prime number $p$ and an $N \times N$ matrix $A = (A_{i,j})$ ($1\leq i,j\leq N$). Each element of $A$ is an integer between $0$ and $p-1$, inclusive. Consider a matrix $B$ obtained by replacing each zero in $A$ with an integer between $1$ and $p-1$, inclusive. There are $(p-1)^K$ such matrices $B$, where $K$ is the number of zeros in $A$. Find each element, modulo $p$, of the sum of $B^p$ over all possible $B$. ## Constraints * $1 \leq N \leq 100$ * $p$ is a prime such that $1 \leq p \leq 10^9$. * $0 \leq A_{i,j} \leq p-1$ * All input values are integers. ## Input The input is given from Standard Input in the following format: $N$ $p$ $A_{1,1}$ $\cdots$ $A_{1,N}$ $\vdots$ $A_{N,1}$ $\cdots$ $A_{N,N}$ [samples]

Samples

Input #1

2 3
0 1
0 2

Output #1

0 2
1 2

$B^p$ for all possible $B$ are as follows:

*   $\begin{pmatrix}1&1 \\ 1&2\end{pmatrix}^3=\begin{pmatrix}5&8 \\ 8&13\end{pmatrix}$
*   $\begin{pmatrix}1&1 \\ 2&2\end{pmatrix}^3=\begin{pmatrix}9&9 \\ 18&18\end{pmatrix}$
*   $\begin{pmatrix}2&1 \\ 1&2\end{pmatrix}^3=\begin{pmatrix}14&13 \\ 13&14\end{pmatrix}$
*   $\begin{pmatrix}2&1 \\ 2&2\end{pmatrix}^3=\begin{pmatrix}20&14 \\ 28&20\end{pmatrix}$

Print each element, modulo $p=3$, of their sum $\begin{pmatrix}48&44 \\ 67&65\end{pmatrix}$.

Input #2

Output #2

1 1 1
1 1 1
1 1 1

$B^p$ for all possible $B$ are as follows:

*   $\begin{pmatrix}1&1&1 \\ 1&1&1 \\ 1&1&1\end{pmatrix}^2=\begin{pmatrix}3&3&3\\3&3&3\\3&3&3\end{pmatrix}$

Print each element, modulo $p=2$, of their sum $\begin{pmatrix}3&3&3\\3&3&3\\3&3&3\end{pmatrix}$.

Input #3

Output #3

API Response (JSON)

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    "name": "Matrix Pow Sum",
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      "content": "You are given a prime number $p$ and an $N \\times N$ matrix $A = (A_{i,j})$ ($1\\leq i,j\\leq N$). Each element of $A$ is an integer between $0$ and $p-1$, inclusive. Consider a matrix $B$ obtained by r",
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      "memory_limit": 262144
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    "difficulty": "None",
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    "sign": "arc190_d"
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    {
      "statement_type": "Markdown",
      "content": "You are given a prime number $p$ and an $N \\times N$ matrix $A = (A_{i,j})$ ($1\\leq i,j\\leq N$). Each element of $A$ is an integer between $0$ and $p-1$, inclusive.\nConsider a matrix $B$ obtained by r...",
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