MAT1503 Oct/Nov 2011 exam paper — questions

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  1. Question 1.1 · Matrices and systems of equations · 4 marks

    Consider the following system of linear equations: 2x + y - 3z = 2; 2x - 3y + z = 10; -2x + y + z = -6. For the system of linear equations 2x + y - 3z = 2, 2x - 3y + z = 10, -2x + y + z = -6, determine whether there exists a value of a such that the triple (a, 1, 3) satisfies all three equations simultaneously. If such a value exists, state what a equals; if no such value exists, give a clear explanation for why not.Show the full question

  2. Question 1.2 · Matrices and systems of equations · 7 marks

    Consider the following system of linear equations: 2x + y - 3z = 2; 2x - 3y + z = 10; -2x + y + z = -6. Using the same system of linear equations 2x + y - 3z = 2, 2x - 3y + z = 10, -2x + y + z = -6, write down its augmented matrix and reduce it to generalized row-echelon form. From this reduced form, determine all solutions of the system.Show the full question

  3. Question 2.1 · Matrices and systems of equations · 9 marks

    Consider the nonhomogeneous system of linear equations: x − 2y + z = 4; y − z = 3; (a² − a − 2)z = a + 1, where a is a parameter. For the nonhomogeneous system x − 2y + z = 4, y − z = 3, (a² − a − 2)z = a + 1, determine the value(s) of the parameter a for which the system has: (a) no solution; (b) exactly one solution; (c) infinitely many solutions.Show the full question

  4. Question 2.2 · Matrices and systems of equations · 2 marks

    Consider the nonhomogeneous system of linear equations: x − 2y + z = 4; y − z = 3; (a² − a − 2)z = a + 1, where a is a parameter. Considering the homogeneous system associated with the given nonhomogeneous system (x − 2y + z = 4, y − z = 3, (a² − a − 2)z = a + 1), determine whether there is any value of a for which this associated homogeneous system has no solution. Give a reason for your answer.Show the full question

  5. Question 2.3 · Matrices and systems of equations · 4 marks

    Consider the nonhomogeneous system of linear equations: x − 2y + z = 4; y − z = 3; (a² − a − 2)z = a + 1, where a is a parameter. For the homogeneous system associated with the given nonhomogeneous system (x − 2y + z = 4, y − z = 3, (a² − a − 2)z = a + 1), find all values of a for which the associated homogeneous system has: (a) only the trivial solution; (b) non-trivial solutions.Show the full question

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