Chapter 12
Notes BASIC MATRIX OPERATIONS Matrix (plural: Matrices) an n x m array of elements element a ij Example 1 a 21 = a 13 = Multiply Matrix by a Scalar Distribute scalar to all elements Addition of Matrices Add elements with identical indices Example 2 Calculate 3C. Example 3 (a) Calculate A+B. (b) Calculate A B. (c) Calculate A+C. Example 4 Calculate 7A 5B.
Example 1 Notes MATRIX MULTIPLICATION AB = C For matrix multiplication, if A is n x r and B is r x m, then C is n x m. HOW? 1) multiply the rows of A with the columns of B 2) add results 3) place in intersection of row/column Example 2 Identity Matrix (I) Square matrix n x n with 1's in the diagonal and 0's as every other element. Fact: AI = IA = A Example 3 Inverse Matrix A 1 If A is a square matrix n x n, then Fact: A 1 A = AA 1 = I Fact: A 1 does not always exist
Example 1 Notes GAUSS JORDAN ELIMINATION A process used to solve systems of equations using matrices. HOW? 1) Write equations as an augmented matrix using only coefficients 2) multiply rows by scalars and subtract multiples of rows 3) continue until identity matrix is obtained Solve the system. Example 2 Solve the system.
Notes INVERSES VIA GAUSS JORDAN ELIMINATION How can we determine the inverse of a matrix? Example 1 Determine the inverse of A.
Notes DETERMINANTS For square (n x n) matrices only. Notations: det A OR A 1x1 Matrix Example 1 Calculate det A. 2x2 Matrix Example 2 Calculate det B. Before we can do a 3x3, we need to learn... Minor (and Minor Matrix) a minor is the determinant of the submatrix obtained by crossing out the entire row and column of a chosen element You can find the minor for every element to create a matrix filled with minors. Example 3 Calculate the minor matrix of B. Cofactor Matrix the same as a minor matrix with the signs of the elements flipped in a checkerboard pattern. Notation: cof A Example 4 Calculate cof B. FACT for any n x n matrix: Example 5 Calculate det B by expansion along the 1st row and along the 2nd column. Example 6 Calculate det C by expansion.
Notes SIMPLEX REGIONS 1 Simplex Line Segment x 1 x 2 2 Simplex Triangle (x 3,y 3 ) (x 1,y 1 ) (x 2,y 2 ) 3 Simplex Tetrahedron (Triangular Pyramid) (x 1,y 1,z 1 ) (x 4,y 4,z 4 ) (x 2,y 2,z 2 ) Example 1 What is the area of a triangle with vertices A(2,2), B(15,8), and C(5,10)? (x 3,y 3,z 3 )
Notes CRAMER'S RULE Another way to solve systems instead of GJE 2x2 Systems 3x3 Systems Example 1 Solve the system.
Notes INVERSE MATRIX VIA DETERMINANTS Another way to determine inverses instead of using GJE. Transpose of a Matrix A T Obtained by using original rows as columns. Example 1 Adjoint Matrix adj A Transpose of the cofactor matrix. adj A = (cof A) T Example 2 Inverse Matrix (using determinants) Note: if det A = 0, then A 1 does not exist. A 1 = adj A det A Example 3 Calculate the inverse using determinants. Example 4 Calculate the inverse using determinants. A 1 = adj A det A
Notes DETERMINANTS VIA ELIMINATION Another way to calculate determinants instead of using minor/cof matrices. Triangular Matrix Upper all elements below main diagonal are 0 Lower all elements above main diagonal are 0 Determinants of Triangular Matrices Example 1 Examples Example 2 Example 3 Fact: The determinant of a triangular matrix is the product of the elements in the main diagonal. You can obtain triangular matrices by using GJE row operations. Fact: Addition/Subtraction row operations do not affect the determinant. (However, multiplication and division do.) Example 4 Calculate the determinant via GJE.
Notes LINEAR TRANSFORMATIONS A matrix M can be an operator that transforms (moves) points and lines. Example 1 Map the point X(3,6) to its image point X' with the linear transformation M. Example 2 Map the line y=2x+3 to its image line y'=mx'+b with the linear transformation M. Example 3 Determine the linear transformation matrix M which maps A(1,0) to A'(2,1) and B(0,1) to B'(2,3).
Notes ROTATION MATRIX A linear transformation matrix R. General Rotation Matrix Example 1 Determine point u' by rotating point u(4,3) counter clockwise 20 using a rotation matrix. Example 2 Using vectors, verify that the angle between u and u' is θ=20.
Notes TRANSFORMING FIGURES A linear transformation (matrix M) maps lines to lines. Thus, under a linear transformation, parallelograms map to parallelograms. Example 1 Map the unit square with the linear transformation M. Fact: The area of the parallelogram is equal to the determinant of the linear transformation matrix. Example 2 Determine the area of the resulting parallelogram in example 1. Fact: The area of any parallelogram is equal to the determinant of the matrix formed by the two generating column vectors. Example 3 Determine the area of the parallelogram generated by the two vectors.