Maths 1a-examples-alg-matrix PDF

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Mathematics IA Worked Examples ALGEBRA: MATRICES AND LINEAR EQUATIONS Produced by the Maths Learning Centre, The University of Adelaide. May 1, 2013 The questions on this page have worked solutions and links to videos on the following pages. Click on the link with each question to go straight to the relevant page.

Questions 1. See Page 4 for worked solutions.       3   0 1 −1 10 0   Let A = . , B = 0 , C = 1 −1 1 and D = − 21 0 12 5 2.5 5 Also let I be the 3 × 3 identity. For each of the following, state if the operation is defined and calculate the result if it is defined. (a) BA (b) AB (c) At D (d) DAt (e) BC (f ) CB (g) B + C (h) (B t + C)(BC + I ) 2. See Page 7 for worked solutions. Show that if A, B are (any) two matrices such that AB and BA are both defined then AB and BA are both square matrices.

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3. See Page 8 for worked solutions. Write down the system corresponding to the aug  of linear equations 3 −8 11 0 mented matrix A = −1 0 8 1 . 2 −8 4 10 Find the elementary row operation that transfers A into B, and then find the inverse elementary row operation that transfers B into A, where:     3 −8 11 0 −1 0 8 1 (b) B =  3 −8 11 0  (a) B = −1 0 8 1 2 −8 4 10  1 −4 2 5  1 −8 27 2 (c) B = −1 0 8 1  2 −8 4 10 4. See Page 10 for worked solutions. Consider the following system of linear equations: x2 − 5x3 = 6 2x1 − 2x2 + 11x3 = −12 4x1 − 3x2 + 17x3 = −18 (a) Use Gauss-Jordan elimination to put the augmented matrix corresponding to this system into reduced row echelon form. (b) State which variables are free and which are basic. (c) Find any solutions of the system. (d) Give a geometric interpretation. 5. See Page 13 for worked solutions. Which of the following are linear combinations of u = (1, 1, 2)t and v = (1, 3, −1)t ? (a) (0, 0, 0)t (b) (1, 1, 1)t (c) (1, −3, 8)t 6. See Page 15 for worked solutions.  2 −4 −2 −3 Find the general solution to Ax = b when A =  6 −12 −2 −7  −1 2 4 3 and also:         0 −1 −7 −2        (b) b = −1 (c) b = −15 (d) b = 2  (a) b = 0 0 2 8 7 

Hint: For parts (b), (c) and (d), it may be helpful to find a particular solution first!

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7. See Page 18 for worked solutions. Let A be a 3 × 5 matrix. (a) Write down the elementary matrices which, when multiplied on the left of A, will perform the following elementary row operations: (i) Subtract 2 times row 1 from row 2; (ii) Interchange row 2 and row 3; (iii) Multiply row 3 by −51. (b) Find the matrix E which, when multiplied on the left of A, will perform all three row operations from part (a) in order. 8. See Page 20 for worked solutions.  3 4 Write A = as a product of elementary matrices. 1 2 9. See Page 21 for worked solutions. 1 2 1  Let B = −2 −3 −1. Find B −1 using row operations. 4 11 −2

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1. Click here to go to question list.    3   10 0 0 1 −1   Let A = 0 , C = 1 −1 1 and D = . − 21 0 1 , B = 5 2.5 5 2 Also let I be the 3 × 3 identity. For each of the following, state if the operation is defined and calculate the result if it is defined. (a) BA (b) AB (c) At D (d) DAt (e) BC (f ) CB (g) B + C (h) (B t + C)(BC + I ) Link to video on YouTube

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2. Click here to go to question list. Show that if A, B are (any) two matrices such that AB and BA are both defined then AB and BA are both square matrices. Link to video on YouTube

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3. Click here to go to question list. Write down the system corresponding to the aug  of linear equations 3 −8 11 0 mented matrix A = −1 0 8 1 . 2 −8 4 10 Find the elementary row operation that transfers A into B, and then find the inverse elementary row operation that transfers B into A, where:     −1 0 8 1 3 −8 11 0 (b) B =  3 −8 11 0  (a) B = −1 0 8 1 2 −8 4 10  1 −4 2 5  1 −8 27 2 (c) B = −1 0 8 1  2 −8 4 10 Link to video on YouTube

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4. Click here to go to question list. Consider the following system of linear equations: x2 − 5x3 = 6 2x1 − 2x2 + 11x3 = −12 4x1 − 3x2 + 17x3 = −18 (a) Use Gauss-Jordan elimination to put the augmented matrix corresponding to this system into reduced row echelon form. (b) State which variables are free and which are basic. (c) Find any solutions of the system. (d) Give a geometric interpretation. Link to video on YouTube

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5. Click here to go to question list. Which of the following are linear combinations of u = (1, 1, 2)t and v = (1, 3, −1)t ? (a) (0, 0, 0)t (b) (1, 1, 1)t (c) (1, −3, 8)t Link to video on YouTube

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6. Click here to go to question list.  2 −4 −2 −3    Find the general solution to Ax = b when A = 6 −12 −2 −7 −1 2 4 3 and also:         0 −1 −7 −2        0 −1 −15 (b) b = (c) b = (d) b = 2  (a) b = 0 2 8 7 Hint: For parts (b), (c) and (d), it may be helpful to find a particular solution first! Link to video on YouTube

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7. Click here to go to question list. Let A be a 3 × 5 matrix. (a) Write down the elementary matrices which, when multiplied on the left of A, will perform the following elementary row operations: (i) Subtract 2 times row 1 from row 2; (ii) Interchange row 2 and row 3; (iii) Multiply row 3 by −51. (b) Find the matrix E which, when multiplied on the left of A, will perform all three row operations from part (a) in order. Link to video on YouTube

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8. Click here to  go to question list. 3 4 Write A = as a product of elementary matrices. 1 2 Link to video on YouTube

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9. Click here to go to question list.  1 2 1 −1   Let B = −2 −3 −1 . Find B using row operations. 4 11 −2 Link to video on YouTube

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10. Let A, B and C be n × n matrices such that AB = I and BC = I. Prove that A, B and C are all invertible. Link to video on YouTube

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