Answer True or False for each statement below. For any False statement, give an

Question

Answer True or False for each statement below. For any False statement, give an example that supports your answer. a. A system of linear equations has either no solution, one unique solution, or an infinite number of solutions. b. For a system of linear equations with 5 variables, if the reduced echelon form of the augmented matrix has 3 nonzero rows, then the system must be consistent and the solution set of the system must have 2 "free" variables. c. A homogeneous (m times n) system of linear equations may have no solution if m > n. d. If the system represented by the matrix equation Ax = b is consistent, then solving the matrix equation is equivalent to showing that b can be written as a linear combination of the columns of A. c. An (n times n) matrix A is nonsingular if and only if the matrix equation Ax = b has infinitely many solutions. f. If the matrix A is singular, then AB = AC does not guarantee that B = C.

Explanation / Answer

a. TURE

b. TURE because The variables corresponding to pivot columns in a matrix are called basic variables and other variables are called free variables.

c. FALSE    reason The homogenous system Ax = 0 always has the solution x = 0. It follows that any homogeneous system of equations is consistent . it does not depend on mxn

d. TURE

e. FLASE reason Let A be a n×n matrix. If detA=0, then the linear system Ax=b has no solution or infinitely many solutions. condition .

f. YES TURE

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