Specialist Mathematics · Unit 3 · Further matrices · Matrix algebra and systems of equations
AI-marked practice questions tied to QCAA mark schemes for this exact LO. Free to start.
Start free practiceFull questions, answers and worked solutions unlock when you start a free practice session.
A system of three linear equations in three variables is reduced to row echelon form, producing the augmented matrix shown. Which geometric object best represents the solution set of this system?
Consider the system of three linear equations: \(x + 2y - z = 4\) \(2x + 4y - 2z = 8\) \(3x + 6y - 3z = k\) For what value of \(k\) does this system have infinitely many solutions? Explain the geometric interpretation of this result.
Three planes in three-dimensional space are defined by the equations x + y + z = 6, 2x + 2y + 2z = 8, and x - y + z = 2. Determine the nature of the solution to this system of equations.
Consider the system of three linear equations: x + 2y - z = 4 2x + 4y - 2z = 8 3x + 6y - 3z = 10 Determine whether this system has a unique solution, no solution, or infinitely many solutions. Justify your answer by describing the geometric interpretation of the system.
A system of three linear equations in three variables is given by: 2x - y + 3z = 7 x + 2y - z = k 5x + py + 5z = 21 where k and p are parameters. (a) Use Gaussian elimination to determine the value of p for which the system has either no solution or infinitely many solutions. (3 marks) (b) For the value of p found in part (a), determine the value of k for which the system has infinitely many solutions. (1 mark) (c) When the system has infinitely many solutions with the values found in parts (a) and (b), state whether the three planes intersect in a line or coincide as a single plane. (1 mark)