How does a 2x2 matrix transform the unit square into a parallelogram?

Algebraically, for a matrix $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$, the determinant is $ad - bc$. Geometrically, $ad$ captures the primary rectangular bounds, while subtracting $bc$ corrects for overlapping triangular regions created by off-diagonal shearing components.

Conditions: Matrix is 2x2; Entries are real numbers

The determinant scales the area of any measurable planar figure, not just squares. The absolute value of the determinant acts as a uniform area scaling factor for all regions under the linear transformation.

Conditions: The transformation is linear.; The figure is measurable and has finite area.; Use ordinary Euclidean area in standard orthonormal coordinates.

The formula for the determinant is $ad - bc$. This scalar value is computed by multiplying the diagonal elements ($a$ and $d$) and subtracting the product of the off-diagonal elements ($b$ and $c$).

Conditions: The matrix is 2x2.; The rows are specified as $(a,b)$ and $(c,d)$.; The entries $a,b,c,d$ are real numbers.

For three-dimensional vectors $u, v, w$, the scalar triple product $u\cdot(v\times w)$ is equal to the determinant of the matrix formed by placing these vectors as columns, denoted as $\det[u\ v\ w]$.

Conditions: The vectors $u, v, w$ are three-dimensional.; The vectors are arranged as columns in the matrix.

No, the determinant is a scalar value that can be positive, zero, or negative. It is explicitly not a matrix absolute value, despite the notation $|A|$ sometimes being used.

Conditions: The object is a square matrix.; The notation $|A|$ or $\det(A)$ is being interpreted.