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Determinant of a Square Matrix 1

الكلية كلية الهندسة     القسم  الهندسة الميكانيكية     المرحلة 1
أستاذ المادة احمد كاظم حسين الحميري       13/07/2018 14:31:13
f the matrix entries are real numbers, the matrix A can be used to represent two linear maps: one that maps the standard basis vectors to the rows of A, and one that maps them to the columns of A. In either case, the images of the basis vectors form a parallelogram that represents the image of the unit square under the mapping. The parallelogram defined by the rows of the above matrix is the one with vertices at (0, 0), (a, b), (a + c, b + d), and (c, d), as shown in the accompanying diagram.

The absolute value of ad ? bc is the area of the parallelogram, and thus represents the scale factor by which areas are transformed by A. (The parallelogram formed by the columns of A is in general a different parallelogram, but since the determinant is symmetric with respect to rows and columns, the area will be the same.)

The absolute value of the determinant together with the sign becomes the oriented area of the parallelogram. The oriented area is the same as the usual area, except that it is negative when the angle from the first to the second vector defining the parallelogram turns in a clockwise direction (which is opposite to the direction one would get for the identity matrix).

To show that ad ? bc is the signed area, one may consider a matrix containing two vectors a = (a, b) and b = (c, d) representing the parallelogram s sides. The signed area can be expressed as |a||b|sin? for the angle ? between the vectors, which is simply base times height, the length of one vector times the perpendicular component of the other. Due to the sine this already is the signed area, yet it may be expressed more conveniently using the cosine of the complementary angle to a perpendicular vector, e.g. a? = (-b, a), such that |a?||b|cos? , which can be determined by the pattern of the scalar product to be equal to ad ? bc:

المادة المعروضة اعلاه هي مدخل الى المحاضرة المرفوعة بواسطة استاذ(ة) المادة . وقد تبدو لك غير متكاملة . حيث يضع استاذ المادة في بعض الاحيان فقط الجزء الاول من المحاضرة من اجل الاطلاع على ما ستقوم بتحميله لاحقا . في نظام التعليم الالكتروني نوفر هذه الخدمة لكي نبقيك على اطلاع حول محتوى الملف الذي ستقوم بتحميله .