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Imagine stretching a flat rubber sheet with a unit square drawn on it. If a matrix transforms that square into a skewed parallelogram, exactly how much did its area grow or shrink?
The answer is a single number called the determinant — the scale factor by which a matrix scales 2D areas.

The 2×2 Formula

For any 2×22 \times 2 matrix M=(abcd)M = \begin{pmatrix} a & b \\ c & d \end{pmatrix}, its determinant is written as det⁡(M)\det(M) or ∣M∣|M|.
We compute it by taking the product of the main diagonal minus the product of the opposing diagonal: det⁡(M)=ad−bc\det(M) = ad - bc
In 1812, French mathematician Augustin-Louis Cauchy formalized the modern term determinant to describe how this value 'determines' the geometric and algebraic properties of a matrix.
Let's calculate one directly: find det⁡(A)\det(A) for A=(4123)A = \begin{pmatrix} 4 & 1 \\ 2 & 3 \end{pmatrix}.
Applying the rule gives (4)(3)−(1)(2)=12−2=10(4)(3) - (1)(2) = 12 - 2 = 10, meaning matrix AA scales any region's area by a factor of 1010.
Why does this simple diagonal subtraction give the exact area of the transformed shape?

Geometric Meaning and Derivation

When a matrix acts on the standard basis vectors i=(10)\mathbf{i} = \begin{pmatrix} 1 \\ 0 \end{pmatrix} and j=(01)\mathbf{j} = \begin{pmatrix} 0 \\ 1 \end{pmatrix}, they land at column vectors (ac)\begin{pmatrix} a \\ c \end{pmatrix} and (bd)\begin{pmatrix} b \\ d \end{pmatrix}.
These two new vectors form the edges of a parallelogram whose original unit square had an area of exactly 11.