Extra Problems: Chapter 1

Transcription

1 MA131 (Section ): 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 Ax = b has no free variable. Answer False (d) A linear system with more unknowns than equations always has infinitely many solutions. Answer False (e) Every homogeneous linear system has at least one solution. (f) If A is upper triangular and B ij is the matrix that results when the ith row and jth column of A are deleted, then B ij is upper triangular if i < j. Answer True (g) IfA = [a ij ]isn nmatrixsuchthata T = A, thena jj = 0forj = 1,2,...,n. Answer True (h) If A is upper triangular, then A T is lower triangular. (i) If A and B are matrices such that AB and BA are both defined, then A and B must be square matrices. (j) If A and B are invertible n n matrices and AB = BA, then A 1 B = BA 1. (k) If A is an invertible n n matrix and if r is a real number such that r 0, then (ra) 1 = ra 1. (l) If the product AB is defined and AB = 0, then A or B is zero matrix. (m) If A and B are invertible, then AB T is also invertible. (n) If A is an invertible matrix and B is row equivalent to A, then B is also invertible. Answer True (o) If R is reduced row-echelon form of A, then there is an invertible matrix B such that BA = R. (p) 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 (q) The product of two elementary matrices is an elementary matrix. (r) A is invertible if and only if the reduced row-echelon from of A is identity matrix.

2 MA131 (Section ): Prepared by Asst.Prof.Dr.Archara Pacheenburawana 2 (s) Let A be an n n matrix and let x and y be vectors in R n. If Ax = Ay and x y, then the matrix A must not be invertible. Answer True (t) 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. (a) Determine the value(s) of h such that the matrix is the augmented matrix of a consistent linear system. [ ] 1 3 h Answer h = 5/2 (b) Which of the following matrices are in reduced row-echelon form? (i) (ii) (iii) (iv) Answer (i) and (iii) (c) Which of the following matrices are in reduced row-echelon form? (i) 0 0 (ii) (iii) (iv) Answer (i), (ii), and (iv) (d) Which of the following matrices are in reduced row-echelon form? (i) (ii) (iii) Answer (ii) and (iii) (e) Determine whether the following matrices are in row-echelon form, reduced row-echelon form, both, or neither A = B = (c) C = D = E = (c) F = (f) Which of the following matrices have reduced row-echelon form equal to ?

3 MA131 (Section ): Prepared by Asst.Prof.Dr.Archara Pacheenburawana A = B = C = (g) Let A = Find the reduced row-echelon form of A (h) Let A = Find the reduced row-echelon form of A (i) Consider a linear system whose augmented matrix is of the form β 0 For what values of β will the system have infinitely many solutions? Answer β = 2 (j) Find all values of k for which the system of equation has infinitely many solutions. Answer k = 1 kx+ y = 1 x+ky = 1 (k) Find all values of α for which the system of equation 1 α 0 x x 2 = 4 α 0 1 x 3 7 has a unique solution. Answer α ±1 (l) For what values of b and g will the system have infinitely many solutions? Answer b = 4, g = 32 2x+by = 16 4x+8y = g (m) Find all h and k such that the system below has no solution. Answer h = 2 and k 8 x+hy = 2 4x+8y = k

4 MA131 (Section ): Prepared by Asst.Prof.Dr.Archara Pacheenburawana 4 (n) Determine the condition of a and b such that the system below is inconsistent. 6x+3y = a 2x y = b (o) Find all values of λ for which the system of equation has nontrivial solutions. Answer λ = 2, 4 (p) Let p(x) = 2x 2 3x+4 and A = [ ] 9 2 Answer 0 13 (λ 3)x+ y = 0 x+(λ 3)y = 0 [ ] 1 2. Find p(a). 0 3 (q) Find the matrix A such that x x+z A y = 0 z y z for all choices of x, y, and z? Answer A = x x z (r) Find the matrix A such that A y = 0 for all choices of x, y, and z? z y +z Answer A = (s) Let A = and B = Find the third row of AB. [ ] Answer (t) Given the following matrices [ ] A = Find tr(c 2 +AA T ). Answer 30 and C = [ ]

5 MA131 (Section ): Prepared by Asst.Prof.Dr.Archara Pacheenburawana 5 (u) Find all values of a and b such that A and B are singular. [ ] [ ] a+b A =, B = a 3b 7 Answer a = 2, b = 1 [ ] 5 1 (v) If A =, then A = (2A T ) 1. [ ] Answer [ ] 2 3 (w) If (3A) 1 =, then find A. 1 1 Answer A = 1 [ ] 1 3 = [ ] 2 1. Use the given information to find 9 5 [ (x) Find all values of x such that A 1 = A when A = Answer x = 4 (y) Find all values of x such that ] [ ] 1 2x 7 = 1 2 [ ] [ ] 3 x. 2 3 Answer x [ = 2 ] 1 3 (z) Given A =. Find the matrix X such that ( ) X 2 5 T 1 +2I 2 = A. [ ] 7 2 Answer X = In each of the following fill the correct answer in the space provided. (a) A square matrix A is symmetric if A T = A. Find a nonzero 3 3 matrix A such that A is symmetric. (b) A square matrix A is symmetric if A T = A. Determine whether ka is symmetric for any scalar k. (c) A square matrix A is called skew-symmetric if A T = A. If A and B are skew-symmetric n n matrices, determine whether A + B is skew-symmetric. Answer Yes. (d) Let A = AX = B. [ Answer X = ] and B = [ ] [ ] Find the matrix X such that (e) Solve the equation C 1 (A X)B 1 = I for X, assuming that A, B, and C are all n n invertible matrices. Answer X = CB A 1 0 [ ] (f) Find a matrix X such that = X

6 MA131 (Section ): Prepared by Asst.Prof.Dr.Archara Pacheenburawana 6 (g) Let A = and B = Find an elementary matrix E such that EA = B Answer E = (h) Find an elementary matrix E such that EA = B, where A = and B = (i) Find an elementary matrix E such that EA = B, where A = and B = Answer (j) Find an elementary matrix E such that EA = B, where A = B = (k) Given A = and B = Find an elementary matrix E such that EA = B. [ ] 2 1 (l) Let A =. Write A as a product of elementary matrices. 6 4 [ ][ ][ ] Answer A = (m) Let E be an elementary matrix such that EA = B, where A = and B = Find E Answer E 1 =

7 MA131 (Section ): Prepared by Asst.Prof.Dr.Archara Pacheenburawana 7 (n) Let E be an elementary matrix such that EA = B, where A = and B = Find E Answer E 1 = c c (o) Find two values of c such that A = c c c is singular. 8 7 c Answer 0 or 2 or (p) Find all values of a 0 such that A = 1 a 0 is noninvertible Answer a = 1 2 [ ] [ ] (q) Find matrix X such that X = [ ] Answer [ ] [ ] (r) If A = and AB =, determine the matrix B [ ] [ ] (s) If A = and AB =, then find B [ ] Answer B = (t) Let A = Evaluate ( 1 A) (u) Given A = and B = Evaluate (AB) (v) Find x, y, and z such that Answer x = 1,y = 3,z = = x 1 1. y z 1

8 MA131 (Section ): Prepared by Asst.Prof.Dr.Archara Pacheenburawana 8 (w) Let a 1 a 2 a A = a 4 a 5 a 6 and A 1 = a 7 a 8 a Solve the following system Answer x = 1, y = 2, z = 1 a 1 x+a 2 y +a 3 z = 1 a 4 x+a 5 y +a 6 z = 3 a 7 x+a 8 y +a 9 z = 2 (x) What conditions must b 1 and b 2 satisfy in order for the system of equations to be consistent? 3x 1 12x 2 = b 1 x 1 + 4x 2 = b 2 4. Solve the following system by Gaussian elimination with back substitution. x 1 +2x 2 +2x 3 +3x 4 = 4 2x 1 +4x 2 + x 3 +3x 4 = 5 3x 1 +6x 2 + x 3 +4x 4 = 7 Answer x 1 = 2 2s t, x 2 = s, x 3 = 1 t, x 4 = t 5. Solve the following system by Gaussian elimination with back substitution. w +x+2y +z = 1 w x y +z = 0 x+ y = 1 w+x +z = 2 6. Solve the given system of equations using Gauss-Jordan elimination. x 1 +2x 2 3x 3 +x 4 = 1 x 1 x 2 +4x 3 x 4 = 6 2x 1 4x 2 +7x 3 x 4 = 1 Answer x 1 = 2 6s, x 2 = 4+s, x 3 = 3 s, x 4 = s 7. Solve the given system of equations using Gauss-Jordan elimination. x 1 +2x 2 + x 3 x 4 = 2 x 1 + x 2 + x 3 = 3 3x 1 +2x 2 +3x 3 2x 4 = 1 Answer x 1 = 1 t, x 2 = 2, x 3 = t, x 4 = 3

9 MA131 (Section ): Prepared by Asst.Prof.Dr.Archara Pacheenburawana 9 8. Solve the given system of equations using Gauss-Jordan elimination. x 1 + x 2 +3x 3 +4x 4 = 12 2x 1 x 2 5x 3 7x 4 = 19 x 1 x 2 +2x 3 +2x 4 = 5 Answer x 1 = 1 3t, x 2 = 2 t, x 3 = 3, x 4 = t 9. Solve the given system of equations using Gauss-Jordan elimination. x 1 +2x 2 3x 3 + x 4 = 2 3x 1 x 2 2x 3 4x 4 = 1 2x 1 +3x 2 5x 3 + x 4 = 3 Answer x 1 = s+t, x 2 = 1+s t, x 3 = s, x 4 = t 10. Solve the given system of equations using Gauss-Jordan elimination. 2x 1 x 2 x 4 = 0 x 2 2x 3 +x 4 = 1 4x 1 + x 2 6x 3 +x 4 = 3 Answer x 1 = 1 +s, x 2 2 = 1+2s t, x 3 = s, x 4 = t 11. Solve the following system by Gauss-Jordan elimination. x 1 +2x 2 x 3 +3x 4 + x 5 = 2 x 1 2x 2 + x 3 x 4 +3x 5 = 4 2x 1 +4x 2 2x 3 +6x 4 +3x 5 = 6 Answer x 1 = 3 2s+t, x 2 = s, x 3 = t, x 4 = 1, x 5 = Solve the given system of equations using Gauss-Jordan elimination. x 1 3x 2 +4x 3 3x 4 +2x 5 = 5 3x 1 7x 2 +8x 3 5x 4 +8x 5 = 9 3x 2 6x 3 +6x 4 +4x 5 = 5 Answer x 1 = 24+2s 3t, x 2 = 7+2s 2t, x 3 = s, x 4 = t, x 5 = Solve the given system of equations using Gauss-Jordan elimination. x 1 +2x 2 +3x 3 + 4x 4 = 5 2x 1 +5x 2 +7x 3 +11x 4 = 12 x 2 + x 3 + 4x 4 = 3 Answer x 1 = 3 t, x 2 = 1 t, x 3 = t, x 4 = 1

10 MA131 (Section ): Prepared by Asst.Prof.Dr.Archara Pacheenburawana Solve the following system by Gauss-Jordan elimination. 15. Consider the system of equations x 1 +x 2 +x 3 x 4 = 4 x 1 x 2 x 3 x 4 = 2 x 1 +x 2 x 3 +x 4 = 2 x+3y + z = α 2x+βy + z = 1 2x+3y +4z = 3 (a) What conditions must α and β satisfy in order for i. the system has exactly one solution ii. the system has no solutions. iii. the system has infinitely many solutions. (b) Solve this system when α = 0 and β = A square matrix A is symmetric if A T = A. Show that A+A T is symmetric. 17. Let A = and R be the reduced row-echelon form of A. Find the elementary matrices E 1, E 2,..., E k such that E k E k 1 E 2 E 1 A = R. 18. Let A = Find the elementary matrices E 1, E 2, E 3, and E 4 such that E 4 E 3 E 2 E 1 A = A r where A r is the reduced row-echelon form of A Answer E 1 = 1 1 0, E 2 = 0 1 0, E 3 = A r = , E 4 = 0 0 1

11 MA131 (Section ): Prepared by Asst.Prof.Dr.Archara Pacheenburawana Let A = Find the elementary matrices E 1, E 2, E 3, and E 4 such that E 4 E 3 E 2 E 1 A = A r where A r is the reduced row-echelon form of A Let A = Find the elementary matrices E 1, E 2, E 3, and E 4 such that Let E 4 E 3 E 2 E 1 A = A R where A R is the reduced row-echelon form of A A = Find the elementary matrices E 1, E 2,..., E k 1, E k such that E k E k 1 E 2 E 1 A = R where R is the reduced row-echelon form of A Answer E 1 = 1 0 0, E 2 = 0 1 0, E 3 = , E 4 = R = Let A = Find the elementary matrices E 1,E 2,E 3 and E 4 such that E 4 E 3 E 2 E 1 A = R where R is the reduced row-echelon form of A Answer E 1 = 1 0 0, E 2 = 0 1 0, E 3 = 0 1 0, E 4 = 0 1 0, R =

12 MA131 (Section ): Prepared by Asst.Prof.Dr.Archara Pacheenburawana Let A = Find the elementary matrices E 1,E 2,E 3 and E 4 such that E 4 E 3 E 2 E 1 A = R where R is the reduced row-echelon form of A Answer E 1 = 2 1 0, E 2 = 0 0 1, E 3 = 0 1 0, E 4 = 0 1 2, R = Let A = Find the elementary matrices E 1,E 2,E 3 and E 4 such that E 4 E 3 E 2 E 1 A = R where R is the reduced row-echelon form of A Answer E 1 = 0 1 0, E 2 = 0 0 1, E 3 = 0 1 0, E 4 = 0 1 0, R = Let A = Find the elementary matrices E 1,E 2,E 3 and E 4 such that E 4 E 3 E 2 E 1 A = R where R is the reduced row-echelon form of A Answer E 1 = 1 0 0, E 2 = 0 1 0, E 3 = 0 1 0, E 4 = 0 1 1, R =

13 MA131 (Section ): Prepared by Asst.Prof.Dr.Archara Pacheenburawana Let A = Find the elementary matrices E 1, E 2, E 3, and E 4 such that E 4 E 3 E 2 E 1 A = R where R is the reduced row-echelon form of A Given a matrix A = Find (a) A 1. (b) Elementary matrices whose the product is A. 28. Find the inverse of A by using Gauss-Jordan elimination, where A = Find A 1 by any method of your choice when A = Let A = Find A 1 by using Gauss-Jordan elimination Let A = Find A 1 by Gauss-Jordan elimination Let Solve the following system a 1 a 2 a A = a 4 a 5 a 6 and A 1 = a 7 a 8 a a 1 x+a 2 y +a 3 z = 2 a 4 x+a 5 y +a 6 z = 1 a 7 x+a 8 y +a 9 z = Consider the system of linear equations Ax = b where x 1 5 A = 1 1 0, x = x 2, and b = x 3 5

14 MA131 (Section ): Prepared by Asst.Prof.Dr.Archara Pacheenburawana 14 (a) Find A 1 by using Gauss-Jordan elimination. (b) Solve the system Ax = b using A Answer (a) A 1 = (b) x 1 = 1,x 2 = 2,x 3 = Let A = (a) Find A 1 by using Gauss-Jordan elimination. (b) Solve the following system using A 1. (i) x+ y + z = 3 2x+3y +4z = 2 y +3z = 1 (ii) x+ y + z = 1 2x+3y +4z = 4 y +3z = 2 Answer (a) x 1 = 10, x 2 = 10, x 3 = 3 (b) x 1 = 1, x 2 = 2, x 3 = (a) Find A 1 by using Gauss-Jordan elimination when A = (b) Solve the following systems using matrix inversion. (i) x 1 + x 2 + x 3 = 2 x 1 +2x 2 +2x 3 = 1 x 1 +2x 2 +3x 3 = 1 Answer (a) A 1 = (ii) x 1 + x 2 + x 3 = 1 x 1 +2x 2 +2x 3 = 1 x 1 +2x 2 +3x 3 = 0 (b) (i) x 1 = 5, x 2 = 5, x 3 = 2 (ii) x 1 = 3, x 2 = 3, x 3 = Let A = (a) Find A 1 by using Gauss-Jordan elimination. (b) Find the elementary matrices E 1,E 2,E 3 and E 4 such that E 4 E 3 E 2 E 1 A = I Solve the following two systems of linear equations by applying the method of Gauss-Jordan elimination. (a) x 1 x 2 +3x 3 = 8 2x 1 x 2 +4x 3 = 11 x 1 +2x 2 4x 3 = 11 (b) x 1 x 2 +3x 3 = 0 2x 1 x 2 +4x 3 = 1 x 1 +2x 2 4x 3 = 2 Answer (a) x 1 = 1, x 2 = 1, x 3 = 2 (b) x 1 = 0, x 2 = 3, x 3 = 1

15 MA131 (Section ): Prepared by Asst.Prof.Dr.Archara Pacheenburawana Solve the following two systems of linear equations by applying the method of Gauss-Jordan elimination. (a) y +z = 3 x y z = 0 x z = 3 (b) y +z = 6 x y z = 0 x z = Solve the following two systems of linear equations by applying the method of Gauss-Jordan elimination. (a) x y +2z = 3 x+2y z = 3 2y 2z = 1 (b) x y +2z = 1 x+2y z = 4 2y 2z = 2 Answer (a) No solution (b) x 1 = 2 s, x 2 = 1+s, x 3 = s 40. Use Gauss-Jordan method to solve the following systems of linear equations w 42 (a) x y = z w 28 (b) x y = z Solve the following systems using matrix inversion. (Hint: First, find A 1.) (a) x 1 +x 3 = 2 x 1 x 2 = 1 2x 2 +x 3 = 1 (b) x 1 +x 3 = 1 x 1 x 2 = 3 2x 2 +x 3 = Solve the following systems using matrix inversion. (a) x 1 + x 3 = 1 3x 1 +3x 2 +4x 3 = 0 2x 1 +2x 2 +3x 3 = 1 (b) x 1 + x 3 = 3 3x 1 +3x 2 +4x 3 = 2 2x 1 +2x 2 +3x 3 = 1 Answer (a) x 1 = 4, x 2 = 0, x 3 = 3 (b) x 1 = 4, x 2 = 6, x 3 = Solve the following systems using matrix inversion. (a) x 1 +2x 2 x 3 = 1 x 2 + x 3 = 2 x 1 +4x 2 2x 3 = 1 (b) x 1 +2x 2 x 3 = 0 x 2 + x 3 = 3 x 1 +4x 2 2x 3 = 2 Answer (a) x 1 = 1, x 2 = 2, x 3 = 4 (b) x 1 = 2, x 2 = 1, x 3 = 4

16 MA131 (Section ): Prepared by Asst.Prof.Dr.Archara Pacheenburawana Solve the following systems using matrix inversion. (a) x 1 +2x 2 +3x 3 = 2 x 1 +3x 2 +2x 3 = 1 2x 1 +4x 2 +7x 3 = 3 (b) x 1 +2x 2 +3x 3 = 1 x 1 +3x 2 +2x 3 = 0 2x 1 +4x 2 +7x 3 = 2 Answer (a) x 1 = 13, x 2 = 4, x 3 = 1 (b) x 1 = 23, x 2 = 5, x 3 = Solve the following systems using matrix inversion. (a) x 1 x 2 = 2 x 1 +x 2 +x 3 = 1 x 2 +x 3 = 3 (b) x 1 x 2 = 1 x 1 +x 2 +x 3 = 2 x 2 +x 3 = 2 Answer (a) x 1 = 8, x 2 = 6, x 3 = 3 (b) x 1 = 2, x 2 = 1, x 3 = Express A = as LU-decomposition Find an LU-decomposition of A = Answer A = Find an LU-decomposition of A = Answer A = Find an LU-decomposition of A = Answer A =

17 MA131 (Section ): Prepared by Asst.Prof.Dr.Archara Pacheenburawana Find an LU-decomposition of A = Answer A = Find an LU-decomposition of A = Answer A = Find an LU-decomposition of A = Answer A = / / Find an LU-decomposition of A = Answer A = Solve the following system of linear equations by using an LU-decomposition. 2x 1 + x 2 x 3 = 1 4x 1 + x 2 4x 3 = 2 2x 1 x 2 2x 3 = Solve the following system of linear equations by using an LU-decomposition. x+4y +2z +3w = 2 x+2y + z = 2 z +4w = 2 2x+6y +3z + w = 2

18 MA131 (Section ): Prepared by Asst.Prof.Dr.Archara Pacheenburawana Solve the following system of linear equations by using an LU-decomposition. 2x 1 +3x 2 + 4x 3 = 1 6x 1 +8x 2 +10x 3 = 4 2x 1 4x 2 3x 3 = The LU-decomposition of a matrix A is given as follows: L = and U = (a) Write down the sequence of row operations which takes A to U when performing Gaussian elimination. x 1 1 (b) Solve the system Ax = b where x = x 2,b = 2 by using an LUdecomposition. x 3 3 Answer (a) r 2 +2r 1, r 3 3r 1, r 3 r 2, 1 2 r 3 (b) x 1 = 9, x 2 = 8, x 3 = 2