Consistency conditions for systems

3 You are given the matrix \(\mathbf { A } = \left( \begin{array} { r r r } k & - 7 & 4 \\ 2 & - 2 & 3 \\ 1 & - 3 & - 2 \end{array} \right)\).

1. Show that when \(k = 5\) the determinant of \(\mathbf { A }\) is zero. Obtain an expression for the inverse of \(\mathbf { A }\) when \(k \neq 5\).
2. Solve the matrix equation $$\left( \begin{array} { r r r } 4 & - 7 & 4 \\ 2 & - 2 & 3 \\ 1 & - 3 & - 2 \end{array} \right) \left( \begin{array} { l } x \\ y \\ z \end{array} \right) = \left( \begin{array} { c } p \\ 1 \\ 2 \end{array} \right)$$ giving your answer in terms of \(p\).
3. Find the value of \(p\) for which the matrix equation $$\left( \begin{array} { r r r } 5 & - 7 & 4 \\ 2 & - 2 & 3 \\ 1 & - 3 & - 2 \end{array} \right) \left( \begin{array} { c } x \\ y \\ z \end{array} \right) = \left( \begin{array} { c } p \\ 1 \\ 2 \end{array} \right)$$ has a solution. Give the general solution in this case and describe it geometrically.

3

1. Find the value of \(k\) for which the matrix $$\mathbf { M } = \left( \begin{array} { r r r } 1 & - 1 & k \\ 5 & 4 & 6 \\ 3 & 2 & 4 \end{array} \right)$$ does not have an inverse.
   Assuming that \(k\) does not take this value, find the inverse of \(\mathbf { M }\) in terms of \(k\).
2. In the case \(k = 3\), evaluate $$\mathbf { M } \left( \begin{array} { r } - 3 \\ 3 \\ 1 \end{array} \right)$$
3. State the significance of what you have found in part (ii).
4. Find the value of \(t\) for which the system of equations $$\begin{array} { r } x - y + 3 z = t \\ 5 x + 4 y + 6 z = 1 \\ 3 x + 2 y + 4 z = 0 \end{array}$$ has solutions. Find the general solution in this case and describe the solution geometrically.

5 Show that if \(a \neq 3\) then the system of equations $$\begin{aligned} & 2 x + 3 y + 4 z = - 5 \\ & 4 x + 5 y - z = 5 a + 15 \\ & 6 x + 8 y + a z = b - 2 a + 21 \end{aligned}$$ has a unique solution. Given that \(a = 3\), find the value of \(b\) for which the equations are consistent.

7 In the following set of simultaneous equations, \(a\) and \(b\) are constants. $$\begin{aligned} 3 x + 2 y - z & = 5 \\ 2 x - 4 y + 7 z & = 60 \\ a x + 20 y - 25 z & = b \end{aligned}$$

1. In the case where \(a = 10\), solve the simultaneous equations, giving your solution in terms of \(b\).
2. Determine the value of \(a\) for which there is no unique solution for \(x , y\) and \(z\).
3. (a) Find the values of \(\alpha\) and \(\beta\) for which \(\alpha ( 2 y - z ) + \beta ( - 4 y + 7 z ) = 20 y - 25 z\) for any \(y\) and \(z\).
   (b) Hence, for the case where there is no unique solution for \(x , y\) and \(z\), determine the value of \(b\) for which there is an infinite number of solutions.
   (c) When \(a\) takes the value in part (ii) and \(b\) takes the value in part (iii)(b) describe the geometrical arrangement of the planes represented by the three equations.

8

1. Find the solution to the following simultaneous equations. $$\begin{array} { r r r } x + y + & z = & 3 \\ 2 x + 4 y + 5 z = & 9 \\ 7 x + 11 y + 12 z = & 20 \end{array}$$
2. Determine the values of \(p\) and \(k\) for which there are an infinity of solutions to the following simultaneous equations. $$\begin{array} { r r r r } x + & y + & z = & 3 \\ 2 x + & 4 y + & 5 z = & 9 \\ 7 x + & 11 y + & p z = & k \end{array}$$

14 The matrix \(\mathbf { M }\) is defined as $$\mathbf { M } = \left[ \begin{array} { c c c } 5 & 2 & 1 \\ 6 & 3 & 2 k + 3 \\ 2 & 1 & 5 \end{array} \right]$$ where \(k\) is a constant. 14

1. Given that \(\mathbf { M }\) is a non-singular matrix, find \(\mathbf { M } ^ { - 1 }\) in terms of \(k\) 14
2. State any restrictions on the value of \(k\) 14
3. Using your answer to part (a), show that the solution to the set of simultaneous equations below is independent of the value of \(k\) $$\begin{array} { r l c c } 5 x + 2 y + c & = & 1 \\ 6 x + 3 y + ( 2 k + 3 ) z & = & 4 k + 3 \\ 2 x + y + 5 z & = & 9 \end{array}$$