Determinant of a Matrix of Order One

Determinant of a Matrix of Order One

The simplest case of a determinant arises when the matrix has only one element. For a matrix of order $1 \times 1$, the determinant is defined in a way that is both natural and consistent with the properties developed for larger matrices.

Definition

Let $A = [a]$ be a matrix of order $1$ — one row and one column, with single entry $a$. The determinant of $A$, denoted $\det(A)$ or $|A|$, is defined to equal $a$ itself:

Why This Definition Makes Sense

The determinant of any square matrix measures a scaling factor (related to area or volume in higher dimensions). For a $1 \times 1$ matrix, the "volume" it scales is just the number line, and the scaling factor is exactly $a$. The sign therefore matters: if $a = 5$ the determinant is $5$; if $a = -3$ it is $-3$.

Examples

1. If $A = [7]$, then $\det(A) = 7$.
2. If $B = [-4]$, then $\det(B) = -4$.
3. If $C = [0]$, then $\det(C) = 0$.

Connection to Notation

Written with vertical bars, $|[a]| = a$, consistent with the notation $|A|$ for the determinant of any square matrix. For a $1 \times 1$ matrix the bars mean the determinant, not the absolute value — the matrix inside the bars tells you which meaning applies. …