ENGG2013 Unit 8 2x2 Determinant Jan, 2011.

Last time Invertible matrix (a.k.a. non-singular matrix) – Represents reversible linear transformation Gauss-Jordan elimination – Algorithm for compute matrix inverse kshumENGG20132

Carl Friedrich Gauss (1777 – 1853) Borned: Braunschweig, Germany. Died: Göttingen, Germany. Great mathematician in the 18 th century. Legacy Gaussian distribution Gaussian elimination Gauss-Jordan elimination Gaussian curvature in differential geometry Proof of the quadratic reciprocity in number theory Construction of 17-gon by straightedge and compass And much more kshumENGG20133

Wilhelm Jordan (1842 –1899) Borned: Ellwangen, Germany. Died: Hanover, Germany. Geodesist. Remembered for: – Surveying in Germany and Africa – His book “Textbook of Geodesy” (Handbuch der Vermessungskunde) popularizes the Gauss-Jordan algorithm kshumENGG

2  2 Determinant Area of parallelogram kshumENGG20135 (a,b) (c,d) Row 1 is the first vector Row 2 is the second vector d b c a

2  2 Determinant Area of parallelogram kshumENGG20136 (3,1) (2,4) Row 1 is the first vector Row 2 is the second vector Area = 10

Outline Computing 2  2 matrix inverse via determinant Properties of determinant Explain why the absolute value of determinant is the area of parallelogram? kshumENGG20137

Determinant of 2  2 matrix Notation: Given a 2  2 matrix we use the notation to stand for the determinant ps – qr. kshumENGG20138 or

Example kshumENGG20139 Determinant of identity matrix is 1 1 1

Example kshumENGG Determinant of a diagonal matrix is the area of a rectangle w h

A formula for the Inverse of a 2x2 matrix Given 2  2 matrix Want to find the inverse of A A formula for A -1 : If det A is nonzero, we have kshumENGG201311

How to compute the inverse of a 2x2 matrix 1.Exchange the two diagonal entries a, d. 2.Take the negative of the two off-diagonal entries b, c. 3.Divide by the determinant. kshumENGG201312

Application in solving equations Solve If we know the inverse of the 2x2 matrix, we can solve the linear system easily. kshumENGG201313

Properties of determinant 1.A matrix and its transpose has the same determinant kshumENGG “The transpose of a matrix” means reflecting the matrix along the diagonal. Row 1 and row 2 become column 1 and column 2, and vice versa. We write A T for the transpose of matrix A. Proof is obvious, because

Transposing does not change area kshumENGG (3,1) (2,4) Area = 10 (3,2) (1,4) Area = 10

Meta-property Any row property of determinant is a column property, and vice versa kshumENGG By property 1, we have:

Properties of determinant 2.If any row or column is zero, then the determinant is 0. kshumENGG Zero area

Properties of determinant 3.If the two columns (or two rows) are constant multiple of each other, the determinant is zero. kshumENGG Zero area

Properties of Determinant 4.If we exchange of the two columns (or two rows), the determinant is multiplied by –1. kshumENGG (2,1) (1,3) The first kind of elementary row operation

Properties of Determinant 5.If we multiply a row (or a column) by a constant c, the value of determinant also increase by a factor of c. kshumENGG The 2nd kind of elementary row operation (0,1) (1,1) (4,4)

Properties of Determinant 6.If we add a constant multiple of a row (column) to the other row (column), the determinant does not change. kshumENGG The 3rd kind of elementary row operation (1,0) (0,1) (3,1)

Properties of Determinant 7.In the linear transformation represented by a 2  2 matrix, the magnitude of determinant measures the area expanding factor. kshumENGG Multiplied by Square with unit area = ad – bc The ratio of area (0,1) (1,1) (1,0)

Properties of Determinant 8.For any 2x2 matrices A and B, we have the following multiplicative property kshumENGG Multiplied by A Square with unit area Parallelogram area = det(A) Expand by a factor of det(A) Parallelogram area = det(AB) = det(A) det(B) Multiplied by B Expand by a factor of det(B) Expand by a factor of det(AB) = det(A) det(B)

Proof of property 8 Let kshumENGG201324

Determinant as area Using the properties in previous pages, we are now ready to show that the absolute value of det(M) is equal to the area of parallelogram whose sides are the two rows of M. We divide the argument into two steps – The two rows are perpendicular (special case). – The two rows are not perpendicular (general case). kshumENGG201325

Determinant as area (I) Suppose that the two rows are perpendicular i.e., ac+bd = 0 (dot product of [a b] and [c d] are zero) Let Want to show that kshumENGG (a,b) (c,d) The trick is to show instead.

Determinant as area (I) kshumENGG By Property 1 Just write down M and M T By Property 8 By the definition of matrix multiplication Because the dot product ac+bd is zero by assumption By the definition of determinant By Pythagoras theorem, are the height and width of the rectangle.

Determinant as area (II) Suppose that the two rows of M are not perpendicular. Idea: “Slide” the parallelogram to a rectangle, while keeping the area unchanged. kshumENGG (a,b) (c,d)

Determinant as Area (II) kshumENGG Decompose [c d] into two components, one is perpendicular to [a b], and the other along the same direction as [a b]. (a,b) (c,d) Find the constant k (hoemwork exercise) Perpendicular to [a b] (The height of parallelogram) In the same direction as [a b]

Determinant as area (II) Choose a constant k such that Let [c’ d’] is perpendicular to [a b] by our choice of k. By definition the area of parallelogram is equal to But kshumENGG By property 6 By the first part of our proof in p.27

We have proved the following theorem (see the picture in p.5) Theorem: For 2x2 matrix M, the absolute value of det(M) is equal to the area of the parallelogram whose sides are the two rows (or the two columns) of M. kshumENGG201331