Remove All Ones With Row and Column Flips

Remove All Ones With Row and Column Flips

What is this problem about?

The Remove All Ones With Row and Column Flips problem asks whether you can make all entries of a binary matrix zero by flipping entire rows or columns. This medium coding problem has an elegant mathematical insight: it's possible if and only if all rows are identical or pairwise complementary (one row's complement equals another). The array, matrix, math, and bit manipulation interview pattern is demonstrated.

Why is this asked in interviews?

Google asks this because the key insight — that valid flip sequences require all rows to be identical or complements of each other — is non-obvious and tests mathematical reasoning about flip operations.

Algorithmic pattern used

Row normalization + set uniqueness check. Normalize each row: if the first element is 1, flip the entire row (XOR with 1). After normalization, all valid rows should be identical. Check if all normalized rows are the same.

The insight: flipping rows and columns can zero the matrix iff all rows are the same row or its complement. Normalizing to "starts with 0" makes all valid rows identical.

Example explanation

matrix=[[0,1,0],[1,0,1],[0,1,0]]. Normalize (flip rows starting with 1):

• Row 0: [0,1,0] (starts with 0, keep).
• Row 1: [1,0,1] → flip → [0,1,0].
• Row 2: [0,1,0] (keep). All normalized: [0,1,0]. All same → true.

matrix=[[0,1],[1,1]]. Normalize: [0,1],[0,0]. Different → false.

Common mistakes candidates make

• Not normalizing by first element before comparison.
• Comparing raw rows without normalization (misses complementary rows).
• Off-by-one in normalization XOR.
• Not handling empty matrix edge case.

Interview preparation tip

Remove All Ones With Row and Column Flips reveals that bit manipulation problems often have elegant normalization properties. The key: flip operations are their own inverses, so "what can be achieved" = "what's invariant under flips." Normalizing rows to start with 0 makes the invariant visible. Practice similar "flip invariant" problems: "make all rows same by column flips," "minimum flips to match target."