Lesson U2.1 Study Guide Sunday, June 3, 2018 2:05 PM Matrix operations, The Inverse of a Matrix and Matrix Factorization Reading: Lay, Sections 2.1, 2.2, 2.3 and 2.5 (about 24 pages). MyMathLab: Lesson U2.1 Learning Objectives: Basic Compute sums, products, and scalar products of matrices. Compute matrix transposes and products of transposed matrices. Find the inverse of a matrix. Use the inverse of a matrix to solve linear systems. Factor matrices using LU factorization, and use this to solve equations. Advanced Demonstrate understanding of theorems and properties about matrix operations and transposes. Prove theorems and demonstrate concept knowledge about the invertability of matrices. Demonstrate understanding of concepts and theorems about matrix invertability. Relate the inverse of a transformation to the inverse of its standard matrix. Section 2.1 Matrix Operations Video over highlights the main ideas in this section but does not provide any worked examples. This material is foundational but not particularly difficult to understand. Matrix arithmetic is exactly as you would expect it to be. However, matrix multiplication is not. Pay close attention both to its definition and two strategies for calculating it. The best way to understand matrix multiplication is through practice. The transpose of a matrix should be new, but is again rather easy to understand and compute, thought its value will not become apparent till later. Here are some Key definitions that I used in the video. Square Matrices An matrix is called a square matrix Diagonal Matrices is diagonal, if whenever Examples: The Identity Matrix where is called the identity matrix. Example: The column vector of is Matrix arithmetic Matrix equality: Matrices and are equal if for each and. Scalar Multiplication (Scaling): any scalar, is a matrix with. Study Guides Page 1
Matrix Addition (and subtraction) is the matrix whose entries are for all and is the zero matrix, where all entries are zero. all : Arithmetic rules for Matrices Matrices and scalars Matrix multiplication The product is defined only when the number of columns in is equal to the number of rows in, in which case and are called conformable matrices. The product AB will have the same number of rows as A and the same number of columns of B is undefined because they are not conformable Definition If is an matrix and if is a matrix with columns, then the product is the matrix whose columns are: Row-Column Rule for computing AB Warning: Study Guides Page 2
In general In general In general does not imply that either or Algebraic rules for Matrix Multiplication The Transpose of a matrix Matrix Powers any square matrix, Matrix Powers has the interpretation you would expect: Exampels: Section 2.2 The Inverse of Matrix The video overview explains the key concepts of matrix inverse and includes a worked example of matrix A. You should watch the video and then read the section in the ebook. for Here are the key definitions and theorems used in the video. An matrix is nonsingular or invertible if there is a such that: is the inverse of, denoted and for a given, there is only one Study Guides Page 3
A singular matrix does not have an inverse If and are nonsingular then is nonsingular. and, if then Finding the Inverse of a matrix: If then the matrix is not invertible (singular) Theorem If is an invertible matrix, then is invertible If and are invertible matrices then AB is also invertible and If A is an invertible matrix then so is and Elementary Matrices Elementary matrices are formed by performing a single elementary row operation on an identity matrix. Type I: Interchange two rows (switch row 1 and 2) Type II: Multiply a row by a nonzero constant (multiply row 3 by 3) Type III: adding a multiple of 1 row to another row (add 3 times row 3 to row 1) If is an matrix, premultiplying by, that is, performs the row operation on A that was used to generate If is an elementary matrix, is invertible and is an elementary matrix of the same type. Theorem An matrix A is invertible if and only if A is row equivalent to, and in this case any sequence of row operations that reduced also transforms into Study Guides Page 4
NOTE: In the online version of the text there is an error Finding the Inverse of a Matrix Start with Matrix so that the left block is and the right block is Use elimination methods put into the row equivalent form If then is non-singular and Otherwise, is singular (does not have an inverse) Example: find Section 2.3 The over view video for this section is very short, as is the section. The Invertible Matrix Theorem Let be an matrix. Then the following statements are equivalent, that is for a given A either all of the statements are true or all of the statements are false. (Note Taking Note: You should transcribe the statements from the Invertible Matrix Theorem, and as you do make sure you agree with why they must all be true. Invertible Linear Transformations Definition A linear transformation such that: for all (1) And for all (2) is said to be invertible if there exists a transformation Theorem Let be a linear transformation and let A be the standard matrix for. Then is invertible if and only if A is inverible. In that case the linear transformation given by is the unique function satifying (1) and (2) above. Section 2.5 There is not video overview for this section. You should read the section in the textbook. I am providing two Study Guides Page 5
worked examples here because I think that will be more helpful Example 1: Solve the equation using the factorization provided Example 2 Find the LU factorization of A Study Guides Page 6
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