1×1

1×1 commonly denotes either the product of the numbers 1 and 1 or a 1×1 matrix. As a multiplication problem, 1×1 equals 1, reflecting the fact that 1 is the multiplicative identity in arithmetic: any number a satisfies a×1 = a.

In linear algebra, a 1×1 matrix is a matrix with one row and one column, written as [a]. Such a matrix behaves like the scalar a and represents a linear transformation on a one-dimensional vector space, namely a transformation that scales vectors by a. The set of all 1×1 matrices over a field is isomorphic to that field, with matrix operations coinciding with ordinary arithmetic: [a] + [b] = [a + b] and [a][b] = [ab]. The determinant of [a] is a, and the trace of [a] is also a.

Invertibility for a 1×1 matrix is straightforward: [a] has an inverse [1/a] if and only if a ≠ 0. If a = 0, the matrix is singular and not invertible. The eigenvalue of [a] is a, and the matrix acts as scalar multiplication by a on vectors in the space.

Beyond pure math, 1×1 appears in contexts such as modular arithmetic, where 1×1 ≡ 1 (mod n) holds for any modulus n. In all cases, a 1×1 object is effectively a single scalar, whether viewed as a number or as a one-entry matrix, and its algebra mirrors that scalar’s properties.