extraproblem131-ch1.pdf

extraproblem131-ch1.pdf - MA131(Section 750002 Prepared by...

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MA131 (Section 750002): Prepared by Asst.Prof.Dr.Archara Pacheenburawana 1 Extra Problems: Chapter 1 1. In each of the following answer true if the statement is always true and false otherwise in the space provided. (a) If the reduced row-echelon form of the augmented matrix for a linear system has a row of zeros, then the system must have infinitely many solutions. Answer False (b) If A is in row-echelon form, then [ A | b ] is in row-echelon form. Answer False (c) If A is a square matrix, then the system A x = b has no free variable. Answer False (d) A linear system with more unknowns than equations always has infinitely many solutions. Answer False (e) If A is upper triangular and B ij is the matrix that results when the i th row and j th column of A are deleted, then B ij is upper triangular if i < j . Answer True (f) If A = [ a ij ] is n × n matrix such that A T = - A , then a jj = 0 for j = 1 , 2 , . . . , n . Answer True (g) If A is upper triangular, then A T is lower triangular. (h) If the product AB is defined and AB = 0, then A or B is zero matrix. (i) If A and B are invertible, then AB T is also invertible. (j) If A is an invertible matrix and B is row equivalent to A , then B is also invertible. Answer True (k) If R is reduced row-echelon form of A , then there is an invertible matrix B such that BA = R . (l) If A and B are m × n matrices, then B is row equivalent to A if and only if A and B have the same reduced row-echelon form. Answer True (m) The product of two elementary matrices is an elementary matrix. (n) A is invertible if and only if the reduced row-echelon from of A is identity matrix. (o) Let A be an n × n matrix and let x and y be vectors in R n . If A x = A y and x negationslash = y , then the matrix A must not be invertible. Answer True (p) If A has an LU factorization, then the LU factors are uniquely determined. 2. In each of the following fill the correct answer in the space provided.
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MA131 (Section 750002): Prepared by Asst.Prof.Dr.Archara Pacheenburawana 2 (a) Determine the value(s) of h such that the matrix is the augmented matrix of a consistent linear system. bracketleftbigg 1 - 3 h - 2 6 - 5 bracketrightbigg Answer h = 5 / 2 (b) Which of the following matrices are in reduced row-echelon form? (i) 0 1 0 0 0 1 0 0 0 (ii) 1 0 0 0 0 0 0 0 1 (iii) 1 0 0 0 0 1 0 0 0 (iv) 1 1 0 0 1 0 0 0 0 Answer (i) and (iii) (c) Which of the following matrices are in reduced row-echelon form? (i) 0 1 0 0 0 0 (ii) 1 0 1 0 0 0 0 0 1 (iii) 1 0 0 1 0 0 1 1 0 0 0 0 (iv) 1 0 2 0 3 0 1 2 0 3 0 0 0 1 2 Answer (i), (ii), and (iv) (d) Which of the following matrices are in reduced row-echelon form? (i) 1 2 3 0 0 4 0 1 0 (ii) 0 0 0 0 0 1 0 0 0 0 0 1 (iii) 1 2 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 Answer (ii) and (iii) (e) Determine whether the following matrices are in row-echelon form, reduced row-echelon form, both, or neither.
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