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2 by 2 Determinants
In linear algebra, the determinant of a two-by-two matrix is a useful quantity. Mostly it is used to calculate the area of the given quadilateral (convex polygons only) and is also an easy representation of a quadilateral(convex polygons only) to be stored in computers as arrays. Scientists, engineers, and mathematicians use determinants in many everyday applications including image and graphic processing.
Calculating the determinant of a square two-by-two matrix is simple, and is the basis of the Laplace formula used for calculating determinants for larger square matrices.
Given a matrix A, the determinant of A (written as |A|) is given by the following equation:
Properties of (2x2) determinants
The rows and vectors of a 2 by 2 matrix can be associated with points on a cartesian plane, such that each row forms a 2D vector. These two vectors form a parallelogram, as shown in the image below. PROOF: Let the vectors be M(a,b),N(c,d) originating from origin in a 2-D plane with an angle (theta>0) between them(head of one vector touching tail of another vector). But in here it doesn't matter because sin(theta)=sin(2(pi)-theta). Then the other point is P(a+c,b+d). The area of the parallelogram is perpendicular distance from one point say N(c,d) to the base vector, M(a,b) multiplied by the length of the base vector, |M(a,b)|. The parallelogram consists of two triangles hence, the area is two times of a triangle. Let the perpendicular distance be h h=|N(c,d)|* sin(theta(angle between two vectors)) b=|M(a,b)| Area=h * b
The absolute value of the determinant is equal to the area of the parallelogram.
Here is an interesting visual proof of this property.
Note: If the determinant equals zero, there are no solutions (intersections) to the system (aka the lines are parallel).
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