41. Zero Matrix

The zero matrix $0_{𝑚\times 𝑛}$ is the $𝑚\times 𝑛$ matrix with every entry equal to $0$.

The zero vector is the $𝑛\times 1$ 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 $0$ may have different shape than the input $0$.)

41.2. Key properties

• Determinant: $det\left(0_{𝑛\times 𝑛}\right)=0$ — singular
• Rank: $\text{rank}\left(0\right)=0$
• Trace: $\text{tr}\left(0_{𝑛\times 𝑛}\right)=0$
• Null space: $\text{null}\left(0\right)={ℝ}^{𝑛}$ — every vector maps to zero

41.3. Where the zero vector matters

• The zero vector is in every subspace (subspace axiom)
• Linear independence test: $\sum {𝑐}_{𝑖}{𝑣}_{𝑖}=\mathrm{𝟎}$ has only the trivial solution
• Kernel of a linear map: vectors that map to $\mathrm{𝟎}$
• Homogeneous systems: $𝐴𝑥=\mathrm{𝟎}$ — always has at least the trivial solution $𝑥=\mathrm{𝟎}$