Linear Algebra
Class 12

Determinant (2×2 Matrix)

det(abcd)=adbc\det\begin{pmatrix} a & b \\ c & d \end{pmatrix} = ad - bc
−10−50510−30−20−100102030
det(A) = ad - bc

Adjust Variables

Element a
a =
-1010
Element b
b =
-1010
Element c
c =
-1010

Determinant measures "scaling factor" of linear transformation. Zero determinant means transformation collapses dimensions.

Real-World Applications

Linear equations — det(A) ≠ 0 means unique solution exists.

Computer graphics — Area scaling in 2D transformations.

Physics — Cross product magnitude: |a × b| = |det([a; b])|.

Engineering — Stability analysis of systems.

Economics — Leontief model solvability.

Robotics — Manipulator singularity detection.

Machine learning — Covariance matrix invertibility.

Cryptography — Matrix cipher decryption.

Control theory — Controllability and observability.

Geometry — Triangle area from vertex coordinates.

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