Mathematics · Ch 4 — Determinants

Determinant of a Matrix of Order Two

4.2.2

Determinant of a Matrix of Order Two

For a $2 \times 2$ matrix, the determinant is a single number calculated from its four entries. This number has important geometric interpretations (an area scaling factor) and algebraic properties that make it a powerful tool in linear algebra.

Consider a general matrix of order $2 \times 2$ :

A = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}

Its determinant, denoted $\det(A)$ , $|A|$ , or simply $\Delta$ , is defined as:

\det(A) = |A| = \Delta = a_{11}a_{22} - a_{21}a_{12}

The pattern is simple: multiply the entries on the main diagonal (top-left to bottom-right), then subtract the product of the entries on the other diagonal (top-right to bottom-left).

Note

The determinant is defined only for square matrices. For a $2 \times 2$ matrix, the result is always a real number — positive, negative, or zero. …

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