41. Zero Matrix

The zero matrix is the matrix with every entry equal to .

The zero vector is the zero matrix:

41.1. Why it’s “the zero”

It’s the additive identity for matrix addition:

And the absorbing element for matrix multiplication:

(provided shapes are compatible — note the result may have different shape than the input .)

41.2. Key properties

• Determinant: — singular
• Rank:
• Trace:
• Null space: — every vector maps to zero

41.3. Where the zero vector matters

• The zero vector is in every subspace (subspace axiom)
• Linear independence test: has only the trivial solution
• Kernel of a linear map: vectors that map to
• Homogeneous systems: — always has at least the trivial solution