Mathematics

Matrix multiplication follows the same distributive rules as regular numbers, but you have to be careful about order. The key insight is that $0A$ means "zero times each entry of $A$"—and when you multiply any number by zero, you get zero. Since every entry of $A$ gets scaled by zero, every entry in the result becomes zero, giving you the zero matrix. This is why $0A = O$ (the zero matrix), just like $0 \cdot x = 0$ for any real number $x$.

Matrix multiplication follows the same distributive rules as regular numbers, but you have to be careful about order. The key insight is that $0A$ means "zero times each entry of $A$"—and when you multiply any number by zero, you get zero. Since every entry of $A$ gets scaled by zero, every entry in the result becomes zero, giving you the zero matrix. This is why $0A = O$ (the zero matrix), just like $0 \cdot x = 0$ for any real number $x$.