// https://qiita.com/tanakh/items/0ba42c7ca36cd29d0ac8 macro_rules! input { ($($r:tt)*) => { let stdin = std::io::stdin(); let mut bytes = std::io::Read::bytes(std::io::BufReader::new(stdin.lock())); let mut next = move || -> String{ bytes.by_ref().map(|r|r.unwrap() as char) .skip_while(|c|c.is_whitespace()) .take_while(|c|!c.is_whitespace()) .collect() }; input_inner!{next, $($r)*} }; } macro_rules! input_inner { ($next:expr) => {}; ($next:expr,) => {}; ($next:expr, $var:ident : $t:tt $($r:tt)*) => { let $var = read_value!($next, $t); input_inner!{$next $($r)*} }; } macro_rules! read_value { ($next:expr, [ $t:tt ; $len:expr ]) => { (0..$len).map(|_| read_value!($next, $t)).collect::>() }; ($next:expr, $t:ty) => ($next().parse::<$t>().expect("Parse error")); } fn powmod(x: i64, mut e: i64, m: i64) -> i64 { let mut sum = 1; let mut cur = x % m; while e > 0 { if e % 2 != 0 { sum = sum * cur % m; } cur = cur * cur % m; e /= 2; } sum } // p must be prime. // Returns (B, H) where HB = A, rows(B) = rank(A) // A: n * m, B: r * m, H: n * r fn find_basis_mod_p(a: &[Vec], p: i64) -> (Vec>, Vec>) { let n = a.len(); let m = a[0].len(); let aold = a.clone(); let mut a = a.to_vec(); let mut pos = vec![]; let mut r = 0; for i in 0..n { for j in 0..r { let idx = pos[j]; assert_eq!(a[j][idx], 1); let val = a[i][idx]; for k in 0..m { a[i][k] -= a[j][k] * val % p; if a[i][k] < 0 { a[i][k] += p; } } } let mut c = 0; while c < m && a[i][c] == 0 { c += 1; } if c >= m { continue; } a.swap(r, i); pos.push(c); let aic = a[r][c]; let aicinv = powmod(aic, p - 2, p); a[r][c] = 1.into(); for j in c + 1..m { a[r][j] *= aicinv; a[r][j] %= p; } for j in r + 1..n { let ajc = a[j][c]; a[j][c] = 0; for k in c + 1..m { let val = ajc * a[r][k] % p; a[j][k] -= val; if a[j][k] < 0 { a[j][k] += p; } } } r += 1; } let mut tr = vec![vec![0; r]; n]; for i in 0..n { let mut cur = aold[i].clone(); for j in 0..r { let idx = pos[j]; let val = cur[idx]; tr[i][j] = val; for k in idx..m { cur[k] -= val * a[j][k] % p; if cur[k] < 0 { cur[k] += p; } } } assert!(cur.iter().all(|&x| x == 0), "{:?} {:?} {:?}", cur, a, aold); } (a[..r].to_vec(), tr) } fn main() { input! { l: usize, m: usize, n: usize, a: [[i64; m]; l], b: [[i64; n]; m], } let mut c = vec![vec![0; n]; l]; for i in 0..l { for j in 0..m { for k in 0..n { c[i][k] += a[i][j] * b[j][k]; } } } if (0..l).any(|i| (0..n).any(|j| c[i][j] != 0)) { eprintln!("not zero"); println!("No"); return; } let p = 1_000_000_007; let mut a = a; let mut b = b; for v in &mut a { for v in v { if *v < 0 { *v += p; } } } for v in &mut b { for v in v { if *v < 0 { *v += p; } } } let (b, _) = find_basis_mod_p(&b, p); let rb = b.len(); let (a, _) = find_basis_mod_p(&a, p); let ra = a.len(); println!("{}", if m - ra == rb { "Yes" } else { "No" }); }