3×a

3×a denotes the product of the scalar 3 and the quantity a. In standard arithmetic this is equal to 3a, the result of multiplying a by 3. The expression is read as “three times a” and is a form of scalar multiplication.

If a is a real or complex number, 3×a equals 3a, such as 3×5 = 15 or 3×(-2) = -6. The operation is commutative with other scalars and respects usual algebraic rules: 3×(b+c) = 3b + 3c, and (3r)×a = 3×(ra) for any scalar r. The identity element for multiplication satisfies 1×a = a, and the zero element satisfies 0×a = 0 for any a.

In linear algebra, 3×a often denotes scalar multiplication of a vector or matrix. For a vector v, 3×v scales every component by 3, changing its magnitude but not its direction. For a matrix A, 3×A multiplies every entry by 3. Scalar multiplication distributes over matrix addition, so 3×(A+B) = 3A + 3B, and is compatible with matrix multiplication in the sense that (3A)B = 3(AB) and A(3B) = 3(AB) when defined. The determinant of an n×n matrix scales by 3^n under 3×A, as det(3A) = 3^n det(A).

In other contexts, a may be a function or an element of a module or vector space, and 3×a denotes scaling by the integer 3 (viewed as a ring element or scalar). While 3×a and 3a are commonly interchangeable, many texts prefer the compact 3a notation for readability.

See also: scalar multiplication, linear algebra, matrix arithmetic, vector space.