What is the formula for the determinant of a 2x2 matrix with rows (a,b) and (c,d)?

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.

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.

The determinant of a 2x2 matrix represents the signed area scaling factor of the linear transformation it defines. Geometrically, it is the area of the parallelogram formed by the matrix's column vectors, starting from the unit square.

Conditions: The matrix is 2x2.; The determinant is calculated as $ad-bc$ for rows $(a,b)$ and $(c,d)$.; The geometric interpretation assumes the standard Euclidean area measure.