Concept explainers
True or False? In Exercises 85 and 86, determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriaste statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text.
(a) For the product of two matrices to be defined, the number of columns of the first matrix must equal the number of rows of the second matrix.
(b) The system
Trending nowThis is a popular solution!
Chapter 2 Solutions
Bundle: Elementary Linear Algebra, 8th + WebAssign Printed Access Card for Larson's Elementary Linear Algebra, 8th Edition, Single-Term
- True or False? In Exercises 59 and 60, determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text. (a) A 63 matrix has six rows. (b) Every matrix is row-equivalent to a matrix in row- echelon form. (c) If the row- echelon form of the augmented matrix of a system of linear equations contains the row 1 0 0 0 0, then the original system is inconsistent. (d) A homogeneous system of four linear equations in six variables has infinitely many solutions.arrow_forwardTrue or False? In Exercises 41 and 42, determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text. a The zero matrix is an elementary matrix. b A square matrix is nonsingular when it an be written as the product ofelementary matrices. c Ax=O has only the trivial solution if and only if Ax=b has a unique solution for every n1 column matrix b.arrow_forwardTrue or False? In Exercises 41 and 42, determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text. a The identity matrix is an elementary matrix. b If E is an elementary matrix, then 2E is an elementary matrix. c The inverse of an elementary matrix is an elementary matrix.arrow_forward
- True or False In Exercises 85 and 86, determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriaste statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text. a If A is a mn matrix and B is a nr matrix, then the product AB is an mr matrix. b The matrix equation Ax=b where A is the coefficient matrix and x and b are column matrices, can be used to represent a system of linear equations.arrow_forwardTrue or False? In Exercises 73 and 76, determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text. a If an mn matrix B can be obtained from elementary row operations on an mn matrix A, then the column space of B is equal to the column space of A. b The system of linearity equations Ax=b is inconsistent if and only if b is in the column space of A.arrow_forwardTrue or False ? In Exercise 73-76, determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If statement is false, provide an example that shows the statement isnt rue in all case or cites an appropriate statement from the text. a The column space of matrix A is equal to the row space of AT. b The row space of a matrix A is equal to the column space of AT.arrow_forward
- True or False? In Exercises 37 and 38, determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text. a If A is a diagonalizable matrix, then it has n linearly independent eigenvectors. b If an nn matrix A is diagonalizable, then it must have n distinct eigenvalues.arrow_forwardTrue or False ? In Exercises 71 and 72, determine whether each statement is true or false. If a statement is true or false. If a statement is true, give a reason or cite an appropriaste statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text. a The inverse of the inverse of a non-singular matrix A, A-1-1, is equal to A itself. b The matrix abcd is invertible when ab-dc0. c If A is a square matrix, then the system of linear equations Ax=b has a unique solution.arrow_forwardSymmetric and Skew-Symmetric Matrices In Exercises 71-74, determine whether the matrix is symmetric, skew-symmetric, or neither. A square matrix is skew-symmetric when AT=-A. A=2113arrow_forward
- Elementary Linear Algebra (MindTap Course List)AlgebraISBN:9781305658004Author:Ron LarsonPublisher:Cengage Learning