What is determinant?

matrix are mathematical objects that transform shapes. Determinant of square matrix A, marked | and | is a number that summarizes the effect and has a number on size and orientation. If [ and b ] the upper line vector for and [ c d ] and | = ad-bc .

Determinant encodes useful information on how the matrix transforms regions. The absolute value of the determinant refers to a matrix scale, how much the character stretches or decreases. His sign describes whether the matrix turns over and brings a mirror image. Matrix can also distort areas and turn them, but this information is not provided by a determinant.

Arithmetically, the transformation effect of the matrix is ​​determined by multiplication of the matrix. If and the matrix is ​​2 × 2 with the upper line [ and b ] and the lower line [ c d ], then [1 0] * a = [ and b ] and [0 1] * a = [ c dd ]. This means that and the point (1,0) to the point ( a, b ) and point (0.1) to the point ( c, d ). UniversityEchny matrix leaves the origin, so one sees that one transforms a triangle with endpoints into (0.0), (0.1) and (1.0) into another triangle with end points at (0.0), ( a, b ) and ( c, d ). The ratio of this new triangle to the original triangle is equal to ad-BC |, absolute value and |.

The design of the matrix determinant describes whether the matrix turns the shape. In view of the triangle with endpoints on (0.0), (0.1) and (1.0), if the matrix and maintain a point (0.1) stationary, while and took over (-1.0) then overturned | and | and | and | and | and | and | and | and | and | will be negative. Matrix does not change the size of the area so | And | must be -1 to be in accordance with the rule that the absolute value | And | The depths of how much the character is stretched.

Mator arithmetic is monitored by the associative law, which means that ( in *a)*b = in *(a*b). GeometricThis means that the combined effect of the first shape transformation with the matrix A and then transforming the shape of the matrix B is equivalent to the transformation of the original shape with the product (A*B). This observation can be inferred that and |*| B | = | A*B |.

Equation | and | * | B | = | A*B | It has an important consequence when and | = 0. In this case, the action cannot be? = 1. From | and | * | B | = | A * B |, this last observation leads to impossible equation 0 * | B | = 1.

The conversation command can also be shown: if and is a square matrix with non -zero determinants, then and has inverse . Geometrically it is an action of any matrix that does not compare Akra. For example, when climbing the square to the line segment, it can be returned to another matrix called its inverse. Such inverse is the matrix analog of mutual.