Matrix Algebra. Learning Objectives. Size of Matrix

Transcription

1 Matrix Algebra 1 Learning Objectives 1. Find the sum and difference of two matrices 2. Find scalar multiples of a matrix 3. Find the product of two matrices 4. Find the inverse of a matrix 5. Solve a system of linear equations using an inverse matrix 2 Size of Matrix The size of a matrix is m (the number of rows) by the n (the number of columns). If n = m we say it is a square matrix s 4 3 is 2 x is 3x is 2 x

2 s by 2 square matrix 3 2 a11 a12 a13 a21 a22 a23 a31 a32 a 33 2 by 1 matrix 3 by 3 square matrix 4 Special Matrices In the zero matrix all entries are zero An identity matrix is square where the diagonal entries are one and all other entries are zero 1 0 I I Equality of Matrices Two matrices are equal if they have the same order and their corresponding entries are equal. Solve for a11, a12, a21, a22 a 4 11 a 3 a a a11 a a21 a

3 Adding Matrices To add two matrices of the same order, add their corresponding entries Add A B A, B AB Subtracting Matrices To subtract two matrices of the same order, subtract their corresponding entries Subtract A B A, B AB Scalar Multiplication To multiply a matrix by a scalar, multiply each entry by the scalar Multiply ka 4 3 A, k ka

4 Find 2A3B A, B A3B Find 2A3B A, B Properties of Matrices Let A, B, and C be mxn matrices and let c and d be scalars. 1) Commutative Property of Matrix Addition: A + B = B + A AB A, B BA

5 Properties of Matrices Let A, B, and C be mxn matrices and let c and d be scalars. 2) Associative Property of Matrix Addition: A + (B + C) = (A + B) + C A, B, C A B C A B C Properties of Matrices Let A, B, and C be mxn matrices and let c and d be scalars. 3) Associative Property of Scalar Multiplication: (cd)a = c(da) cd A 4 3 A, c 2, d cda Properties of Matrices Let A, B, and C be mxn matrices and let c and d be scalars. 4) Scalar Identity: 1A = A 4 3 A A1 A

6 Properties of Matrices Let A, B, and C be mxn matrices and let c and d be scalars. 5) Distributive Property (two forms): c(a + B) = ca + cb (c + d)a = ca + da 16 s A, B, c 2, d c(a + B) = ca + cb c A B ca cb (c + d)a = ca + da c d A ca da Properties of Matrices Let A, B, and C be mxn matrices and let c and d be scalars. 6) If A is an mxn matrix and O is the mxn zero matrix, then A + O = A 4 3 A A 0 A

7 Vector Vectors can be represented as either row vectors or column vectors. A row vector is a 1 by n matrix 1 2 n R r r r A Column vector is a n by 1 matrix c1 c2 C cn 19 Product of Vectors 1 2 n R r r r R is 1 x n C is n x 1 RC is 1 x 1 RC r r r 1 2 n c1 c2 cn c1 c2 C cn rc 1 1 r2 c2 rnc n 20 R RC C

8 Computing Revenue We sell pizzas for $ pizzas for $ pizzas for $15 Quantity R $5 Cost C $10 $15 $ $10 $15 22 Revenue RC Computing Revenue We sell pizzas for $ pizzas for $ pizzas for $15 $5 RC $10 $ $5 125 $10 90 $15 $500 $10 $1350 $ Multiplying Matrices If A is a m by n matrix and B is a n by p matrix Then the product AB is a m by p matrix mnnp Inside must equal mnnp Outside is product 24 8

9 2 1 0 Find AB A B First check to see size of product Then write A as two row vectors A 2 1 0, A and write B as two row vectors 1 3 B1 1, B A B The product is then AB A B A B A B AB AB AB AB AB Find AB A B

10 2 1 0 Find BA A B We see that AB BA 28 Remarks In the last example AB BA That is matrix multiplication is not commutative. In fact, if A is 2 by 3 and B is 3 by 5, AB is 2 by 5, but BA is not even possible to compute, since the size is wrong. 29 Properties of Matrices Let A be a m by n matrix 1) Associative Property A(BC) = (AB)C 2) Distributive Property A(B+C) = AB+AC (A+B)C = AC+BC 3) Identity I m A = A A I n = A 30 10

11 Application: Computer Graphics A computer image is usually represented as a matrix picture elements (pixels) The number of pixels determines the resolution of the image. Typical resolutions range from 320 by 200 to 2000 by 1500 A gray-scale (black and white) image uses a number (0-255) to describe the intensity of each pixel 31 Application: Computer Graphics If we enlarge the eye of the bird we can see the pixels Each pixel is represented by a number associated with color and brightness 32 Application: Computer Graphics A matrix representation of the letter a 33 11

12 Inverse of Matrix Let A be a m by m square matrix A -1 is an inverse of A if A A -1 = A -1 A = I A B Inverses do not always exist 34 Finding The Inverse We have several methods of finding the inverse suppose A 3 4 and we want to find the 5 6 a b Inverse. We wan to find B such that c d 3 4 a b 1 0 AB I and after 5 6 c d 0 1 3a 4c 3b 4d 1 0 multiplying we have 5a 6c 5b 6d 0 1 Finding The Inverse 3a 4c 3b 4d 1 0 5a 6c 5b 6d 0 1 We then have to solve four equations 3a4c1 5a6c0 and 3b4d 0 5b6d A

13 Finding The Inverse Another way to find the inverse is to begin with the augmented matrix A I which we row reduce to produce which is equal to I A giving us A Finding The Inverse The inverse of a 2x2 matrix can be found by a b Let A c d If ad bc 0, then A is invertible and A 1 1 d b ad bc c a det Remarks If the determinant is zero, we say the matrix is singular and the inverse matrix does not exist a b A ad bc c d 13

14 A A Finding The Inverse The inverse of a matrix can also be found using a graphing calculator A TI

15 7 3 A A Matrix Equation a11x a12 y a13 z b1 a21x a22 y a23z b becomes 2 a31x a32 y a33z b3 where and a a a A a a a a a a x b1 x y b b2 z b 3 Ax = b 44 Matrix Equation Where the inverse exists Ax = b is solved by x = A -1 b Note: This works in the case of consistent independent systems i.e. those with only one solution 45 15

16 3x y z 2 2x y 3z 14 x y z 4 Solution 2, 1,