MAT1503 May/Jun 2011 exam paper — questions

Free sample

  1. Question 1 · Matrices and systems of equations · 8 marks

    A system of linear equations is given: x1 + x2 - x3 = 2; 3x1 + 2x2 - x3 = 3; -x1 - x2 + 2x3 = -1. Write down the augmented matrix representing this system, reduce the augmented matrix to generalized row-echelon form, and hence determine the solution of the system.Show the full question

  2. Question 2.1 · Elementary matrices and inverses · 3 marks

    This question reuses the same system of linear equations given in Question 1, namely x1 + x2 − x3 = 2, 3x1 + 2x2 − x3 = 3, and −x1 − x2 + 2x3 = −1. Suppose this system is written in matrix form as A·x = b, where x is the column vector containing x1, x2 and x3. Write down the coefficient matrix A and the right-hand-side column vector b corresponding to the system x1 + x2 − x3 = 2, 3x1 + 2x2 − x3 = 3, −x1 − x2 + 2x3 = −1, expressed in the matrix form A·x = b.Show the full question

  3. Question 2.2 · Elementary matrices and inverses · 3 marks

    This question reuses the same system of linear equations given in Question 1, namely x1 + x2 − x3 = 2, 3x1 + 2x2 − x3 = 3, and −x1 − x2 + 2x3 = −1. Suppose this system is written in matrix form as A·x = b, where x is the column vector containing x1, x2 and x3. State the respective sizes (dimensions) of the matrices A, x and b that you identified for the system A·x = b corresponding to the equations x1 + x2 − x3 = 2, 3x1 + 2x2 − x3 = 3, −x1 − x2 + 2x3 = −1.Show the full question

  4. Question 2.3 · Elementary matrices and inverses · 8 marks

    This question reuses the same system of linear equations given in Question 1, namely x1 + x2 − x3 = 2, 3x1 + 2x2 − x3 = 3, and −x1 − x2 + 2x3 = −1. Suppose this system is written in matrix form as A·x = b, where x is the column vector containing x1, x2 and x3. Using the matrix inverse algorithm — that is, by performing a sequence of elementary row operations on the augmented matrix [A | I] — determine the inverse matrix A⁻¹ of the coefficient matrix A associated with the system x1 + x2 − x3 = 2, 3x1 + 2x2 − x3 = 3, −x1 − x2 + 2x3 = −1.Show the full question

  5. Question 3 · Matrices and systems of equations · 6 marks

    Using the answers you found in parts 2.1, 2.2 and 2.3, together with the matrix equation Ax = b, determine the solution of the same system of linear equations given in both Question 1 and Question 2 (namely x1+x2-x3=2, 3x1+2x2-x3=3, -x1-x2+2x3=-1), making sure to show every step of your reasoning. Note that if the solution you obtain here does not match the solution you found in Question 1, this indicates a writing or numerical error somewhere, and you should use this check to find and correct it.Show the full question

  6. Question 4.1 · Determinants and Cramer's rule · 7 marks

    Consider the square matrix C = [[1, 1, -1], [3, 0, -1], [-1, -1, 2]]. For the matrix C = [[1, 1, -1], [3, 0, -1], [-1, -1, 2]], compute det(C) by performing a cofactor expansion along the first row only. Show every step of the calculation; no marks will be given if any other expansion method or technique is used.Show the full question

  7. Question 4.2 · Determinants and Cramer's rule · 2 marks

    Consider the square matrix C = [[1, 1, -1], [3, 0, -1], [-1, -1, 2]]. For the same matrix C, consider the homogeneous system Cx = 0, where x = [x1, x2, x3]^T and 0 = [0, 0, 0]^T. State whether this homogeneous system has no solution, exactly one solution, or infinitely many solutions, and justify your answer.Show the full question

  8. Question 4.3(a) · Determinants and Cramer's rule · 2 marks

    Consider the square matrix C = [[1, 1, -1], [3, 0, -1], [-1, -1, 2]]. Using the value of det(C) found in 4.1, and showing your reasoning, evaluate det(C C^-1).Show the full question

  9. Question 4.3(b) · Determinants and Cramer's rule · 2 marks

    Consider the square matrix C = [[1, 1, -1], [3, 0, -1], [-1, -1, 2]]. Using the value of det(C) found in 4.1, and showing your reasoning, evaluate det(3C).Show the full question

  10. Question 4.3(c) · Determinants and Cramer's rule · 3 marks

    Consider the square matrix C = [[1, 1, -1], [3, 0, -1], [-1, -1, 2]]. Using the value of det(C) found in 4.1, and showing your reasoning, evaluate det(C C^T).Show the full question

  11. Question 4.3(d) · Determinants and Cramer's rule · 2 marks

    Consider the square matrix C = [[1, 1, -1], [3, 0, -1], [-1, -1, 2]]. Using the value of det(C) found in 4.1, and showing your reasoning, evaluate det(2C^-1).Show the full question

  12. Question 5.1 · Elementary matrices and inverses · 5 marks

    This question uses six 3×3 matrices, labelled A, B, C, D, F and G, given as follows: A has rows (0,0,1), (0,2,0), (1,0,0); B has rows (1,0,0), (0,3,0), (0,0,1); C has rows (0,0,1), (0,1,0), (1,0,0); D has rows (1,0,0), (-2,1,0), (1,0,0); F has rows (1,0,0), (-2,1,0), (0,0,1); and G has rows (0,1,0), (1,0,0), (0,3,1). Given the six 3×3 matrices A, B, C, D, F and G as described, (a) identify which of these six matrices are elementary matrices, and write down only those that qualify as elementary matrices; and (b) state the general definition of an elementary matrix.Show the full question

  13. Question 5.2 · Elementary matrices and inverses · 4 marks

    This question uses six 3×3 matrices, labelled A, B, C, D, F and G, given as follows: A has rows (0,0,1), (0,2,0), (1,0,0); B has rows (1,0,0), (0,3,0), (0,0,1); C has rows (0,0,1), (0,1,0), (1,0,0); D has rows (1,0,0), (-2,1,0), (1,0,0); F has rows (1,0,0), (-2,1,0), (0,0,1); and G has rows (0,1,0), (1,0,0), (0,3,1). (a) Determine whether the inverse of an elementary matrix is itself also an elementary matrix. (b) Determine the inverse matrices of each of the elementary matrices that you identified in sub-question 5.1(a).Show the full question

  14. Question 6.1(a) · Lines and planes · 10 marks

    Let L1 be the straight line through the points A(3, 0, 2) and B(4, 3, 0), and let L2 be the straight line through the points B(4, 3, 0) and C(8, 1, -1). Determine parametric equations for both L1 and L2.Show the full question

  15. Question 6.1(b) · Lines and planes · 5 marks

    Using the same lines L1 (through A(3, 0, 2) and B(4, 3, 0)) and L2 (through B(4, 3, 0) and C(8, 1, -1)) as in 6.1(a), determine whether L1 and L2 are mutually perpendicular. Show all your working.Show the full question

  16. Question 6.2 · Lines and planes · 3 marks

    Determine, showing all your working, whether the point (8, 4, -5) lies on the plane given by 7x - 3y + 4z = 8.Show the full question

  17. Question 6.3 · Lines and planes · 7 marks

    A line L passes through the points P1(2, 4, -1) and P2(5, 0, 7). Determine the point at which L intersects the xy-plane.Show the full question

  18. Question 7.1 · Lines and planes · 3 marks

    Find an equation for the plane that passes through the point (1, 1, 1) and is parallel to the plane given by x - 3y - 2z - 4 = 0.Show the full question

  19. Question 7.2 · Lines and planes · 3 marks

    Calculate the volume of the parallelepiped in 3-space formed by the three vectors u = (1, 3, -1), v = (1, 1, 2) and w = (3, -1, 2).Show the full question

  20. Question 7.3 · Lines and planes · 4 marks

    Work out the distance between the plane 2x - 3y + 6z = -1 and the point (1, -4, 3).Show the full question

  21. Question 8.1 · Complex numbers and De Moivre's theorem · 4 marks

    Write the complex number 1 + i√3 in polar form.Show the full question

  22. Question 8.2(a) · Complex numbers and De Moivre's theorem · 1 mark

    State De Moivre's theorem.Show the full question

  23. Question 8.2(b) · Complex numbers and De Moivre's theorem · 5 marks

    Apply De Moivre's theorem to determine all the cube roots of −8.Show the full question