Conditions for unique solution

8

1. Find the values of \(a\) for which the system of equations $$\begin{aligned} 3 x + y + z & = 0 \\ a x + 6 y - z & = 0 \\ a y - 2 z & = 0 \end{aligned}$$ does not have a unique solution.
   The matrix \(\mathbf { A }\) is given by $$\mathbf { A } = \left( \begin{array} { r r r } 3 & 1 & 1 \\ 0 & 6 & - 1 \\ 0 & 0 & - 2 \end{array} \right) .$$
2. Use the characteristic equation of \(\mathbf { A }\) to find the inverse of \(\mathbf { A } ^ { 2 }\).
3. Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { A } ^ { 5 } = \mathbf { P D P } ^ { - 1 }\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

1

1. Given that \(a\) is an integer, show that the system of equations $$\begin{aligned} a x + 3 y + z & = 14 \\ 2 x + y + 3 z & = 0 \\ - x + 2 y - 5 z & = 17 \end{aligned}$$ has a unique solution and interpret this situation geometrically.
2. Find the value of \(a\) for which \(x = 1 , y = 4 , z = - 2\) is the solution to the system of equations in part (a).

4 The matrix equation \(\left( \begin{array} { r r } 6 & - 2 \\ - 3 & 1 \end{array} \right) \binom { x } { y } = \binom { a } { b }\) represents two simultaneous linear equations in \(x\) and \(y\).

1. Write down the two equations.
2. Evaluate the determinant of \(\left( \begin{array} { r r } 6 & - 2 \\ - 3 & 1 \end{array} \right)\). What does this value tell you about the solution of the equations in part (i)?

3 By using the determinant of an appropriate matrix, find the values of \(k\) for which the simultaneous equations $$\begin{aligned} & k x + 8 y = 1 \\ & 2 x + k y = 3 \end{aligned}$$ do not have a unique solution.

2 Matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by \(\mathbf { A } = \left( \begin{array} { r r } a & 1 \\ - 1 & 3 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { l l } - 2 & 5 \\ - 1 & 0 \end{array} \right)\) where \(a\) is a constant.

1. Find the following matrices.
   • \(\mathbf { A } + \mathbf { B }\)
   • AB
   • \(\mathbf { A } ^ { 2 }\)
   • 1. Given that the determinant of \(\mathbf { A }\) is 25 find the value of \(a\).
     2. You are given instead that the following system of equations does not have a unique solution.
   $$\begin{array} { r } a x + y = - 2 \\ - x + 3 y = - 6 \end{array}$$ Determine the value of \(a\).

1. A system of three equations is defined by

$$\begin{aligned} k x + 3 y - z & = 3 \\ 3 x - y + z & = - k \\ - 16 x - k y - k z & = k \end{aligned}$$ where \(k\) is a positive constant.
Given that there is no unique solution to all three equations,

1. show that \(k = 2\) Using \(k = 2\)
2. determine whether the three equations are consistent, justifying your answer.
3. Interpret the answer to part (b) geometrically.