Interpreting Determinants

Interpreting Determinants Gautam Sethi

Suppose we have the following 2 × 2 matrix:   a b  G≡ c d Its determinant is written as a b ≡a×d−b×c |G| ≡ c d To interpret what this number represents, consider the parallelogram OYZX drawn in the figure below.

1

The vertex O of the parallelogram is the origin, the coordinates of vertex X are (a, b), the coordinates of vertex Y are (c, d), and the coordinates of vertex Z are (a+c, b+d). The area of parallelogram OYZX can be computed by summing up the areas of its constituent parts: Ar(OY ZX) = Ar(OY T X) + Ar(Y U T ) + Ar(T U ZV ) + Ar(XT V )

(1)

Note that Ar(Y U T ) = Ar(Y ZS) − (Ar(T U ZV ) + Ar(V ZS))

(2)

Ar(XT V ) = Ar(XRZ) − (Ar(T U ZV ) + Ar(U RZ))

(3)

and

Substituting the expressions from equations (2) and (3) in equation (1) and simplifying, Ar(OY ZX) = Ar(OY T X)+Ar(Y ZS)+Ar(XRZ)−(Ar(T U ZV )+Ar(V ZS)+Ar(U RZ)) But Ar(T U ZV ) + Ar(V ZS) + Ar(U RZ) = Ar(T RZS) Therefore, Ar(OY ZX) = Ar(OY T X) + Ar(Y ZS) + Ar(XRZ) − Ar(T RZS) Also note that 4Y ZS ∼ = 4OXP and 4XRZ ∼ = 4OQY . Hence, Ar(OY ZX) = Ar(OY T X) + Ar(OXP ) + ArOQY ) − Ar(T RZS) and that Ar(OY T X) + Ar(OXP ) + ArOQY ) = Ar(OQT P ) Therefore, Ar(OY ZX) = Ar(OQT P ) − Ar(T RZS) Thus, the area of the parallelogram OYZX is identical to the difference between the areas of the rectangles OQTP and TRZS. What are the areas of these rectangles? Clearly Ar(OQT P ) = a × d and Ar(T RZS) = b × c. Therefore, Ar(OY ZX) = a × d − b × c which is exactly the value of the determinant of G! 2