4.03s Consistent/inconsistent: systems of equations

15 The equations of three planes are $$\begin{aligned} - 4 x + k y + 7 z & = 4 \\ x - 2 y + 5 z & = 1 \\ 2 x + 3 y + z & = 2 \end{aligned}$$ Given that the planes form a sheaf, determine the values of \(k\) and \(l\).

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.

7. $$\mathbf { M } = \left( \begin{array} { r r r } 2 & - 1 & 1 \\ 3 & k & 4 \\ 3 & 2 & - 1 \end{array} \right) \quad \text { where } k \text { is a constant }$$

1. Find the values of \(k\) for which the matrix \(\mathbf { M }\) has an inverse.
2. Find, in terms of \(p\), the coordinates of the point where the following planes intersect $$\begin{aligned} & 2 x - y + z = p \\ & 3 x - 6 y + 4 z = 1 \\ & 3 x + 2 y - z = 0 \end{aligned}$$
3. 1. Find the value of \(q\) for which the set of simultaneous equations $$\begin{aligned} & 2 x - y + z = 1 \\ & 3 x - 5 y + 4 z = q \\ & 3 x + 2 y - z = 0 \end{aligned}$$ can be solved.
   2. For this value of \(q\), interpret the solution of the set of simultaneous equations geometrically.

6. $$\mathbf { M } = \left( \begin{array} { r r r } k & 5 & 7 \\ 1 & 1 & 1 \\ 2 & 1 & - 1 \end{array} \right) \quad \text { where } k \text { is a constant }$$

1. Given that \(k \neq 4\), find, in terms of \(k\), the inverse of the matrix \(\mathbf { M }\).
2. Find, in terms of \(p\), the coordinates of the point where the following planes intersect. $$\begin{array} { r } 2 x + 5 y + 7 z = 1 \\ x + y + z = p \\ 2 x + y - z = 2 \end{array}$$
3. 1. Find the value of \(q\) for which the following planes intersect in a straight line. $$\begin{array} { r } 4 x + 5 y + 7 z = 1 \\ x + y + z = q \\ 2 x + y - z = 2 \end{array}$$
   2. For this value of \(q\), determine a vector equation for the line of intersection.

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. $$\begin{array} { r r r r } x + & y + & z = & 3 \\ 2 x + & 4 y + & 5 z = & 9 \\ 7 x + & 11 y + & p z = & k \end{array}$$

7 Three planes have equations $$\begin{aligned} ( 4 k + 1 ) x - 3 y + ( k - 5 ) z & = 3 \\ ( k - 1 ) x + ( 3 - k ) y + 2 z & = 1 \\ 7 x - 3 y + 4 z & = 2 \end{aligned}$$ 7

1. The planes do not meet at a unique point.
   Show that \(k = 4.5\) is one possible value of \(k\), and find the other possible value of \(k\).
   

   7

   For each value of \(k\) found in part (a), identify the configuration of the given planes. In each case fully justify your answer, stating whether or not the equations of the planes form a consistent system. [4 marks] \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)

4 A \(3 \times 3\) system of equations is given by the matrix equation \(\left( \begin{array} { r r r } - 1 & 3 & 1 \\ 5 & - 1 & 2 \\ - 1 & 1 & 0 \end{array} \right) \left( \begin{array} { l } x \\ y \\ z \end{array} \right) = \left( \begin{array} { r } 1 \\ 16 \\ - 2 \end{array} \right)\).

1. Show that this system of equations does not have a unique solution.
2. Solve this system of equations and describe the geometrical significance of the solution.

3

1. Evaluate, in terms of \(k\), the determinant of the matrix \(\left( \begin{array} { c c c } 1 & 2 & 1 \\ - 3 & 5 & 8 \\ 6 & 12 & k \end{array} \right)\). Three planes have equations \(x + 2 y + z = 4 , - 3 x + 5 y + 8 z = 21\) and \(6 x + 12 y + k z = 31\).
2. State the value of \(k\) for which these three planes do not meet at a single point.
3. Find the coordinates of the point of intersection of the three planes when \(k = 7\).

3

1. Evaluate, in terms of \(k\), the determinant of the matrix \(\left( \begin{array} { r r r } 1 & 2 & 1 \\ - 3 & 5 & 8 \\ 6 & 12 & k \end{array} \right)\). Three planes have equations \(x + 2 y + z = 4 , - 3 x + 5 y + 8 z = 21\) and \(6 x + 12 y + k z = 31\).
2. State the value of \(k\) for which these three planes do not meet at a single point.
3. Find the coordinates of the point of intersection of the three planes when \(k = 7\).

9

1. Find the inverse of the matrix \(\left( \begin{array} { r r r } 1 & 3 & 4 \\ 2 & 5 & - 1 \\ 3 & 8 & 2 \end{array} \right)\), and hence solve the set of equations $$\begin{aligned} x + 3 y + 4 z & = - 5 \\ 2 x + 5 y - z & = 10 \\ 3 x + 8 y + 2 z & = 8 \end{aligned}$$
2. Find the value of \(k\) for which the set of equations $$\begin{aligned} x + 3 y + 4 z & = - 5 \\ 2 x + 5 y - z & = 15 \\ 3 x + 8 y + 3 z & = k \end{aligned}$$ is consistent. Find the solution in this case and interpret it geometrically.

Find the rank of the matrix \(\mathbf{A}\), where $$\mathbf{A} = \begin{pmatrix} 1 & 1 & 2 & 3 \\ 4 & 3 & 5 & 10 \\ 6 & 6 & 13 & 13 \\ 14 & 12 & 23 & 45 \end{pmatrix}.$$ [3] Find vectors \(\mathbf{x_0}\) and \(\mathbf{e}\) such that any solution of the equation $$\mathbf{A}\mathbf{x} = \begin{pmatrix} 0 \\ 2 \\ -1 \\ -3 \end{pmatrix} \quad (*)$$ can be expressed in the form \(\mathbf{x_0} + \lambda\mathbf{e}\), where \(\lambda \in \mathbb{R}\). [5] Hence show that there is no vector which satisfies \((*)\) and has all its elements positive. [3]

The linear transformation \(\mathrm{T} : \mathbb{R}^4 \to \mathbb{R}^4\) is represented by the matrix \(\mathbf{M}\), where $$\mathbf{M} = \begin{pmatrix} 1 & -2 & -3 & 1 \\ 3 & -5 & -7 & 7 \\ 5 & -9 & -13 & 9 \\ 7 & -13 & -19 & 11 \end{pmatrix}.$$ Find the rank of \(\mathbf{M}\) and a basis for the null space of \(\mathrm{T}\). [6] The vector \(\begin{pmatrix} 1 \\ 2 \\ 3 \\ 4 \end{pmatrix}\) is denoted by \(\mathbf{e}\). Show that there is a solution of the equation \(\mathbf{M}\mathbf{x} = \mathbf{M}\mathbf{e}\) of the form $$\mathbf{x} = \begin{pmatrix} a \\ b \\ -1 \\ -1 \end{pmatrix}, \text{ where the constants } a \text{ and } b \text{ are to be found.}$$ [4]

The linear transformation \(\mathrm{T} : \mathbb{R}^4 \to \mathbb{R}^4\) is represented by the matrix \(\mathbf{M}\), where $$\mathbf{M} = \begin{pmatrix} 3 & 2 & 0 & 1 \\ 6 & 5 & -1 & 3 \\ 9 & 8 & -2 & 5 \\ -3 & -2 & 0 & -1 \end{pmatrix}.$$

1. Find the rank of \(\mathbf{M}\). [3]

Let \(K\) be the null space of \(\mathrm{T}\).

1. Find a basis for \(K\). [3]
2. Find the general solution of $$\mathbf{M}\mathbf{x} = \begin{pmatrix} 2 \\ 5 \\ 8 \\ -2 \end{pmatrix}.$$ [3]

The matrix \(\mathbf{A}\) is defined by $$\mathbf{A} = \begin{pmatrix} 1 & 5 & 1 \\ 1 & -2 & -2 \\ 2 & 3 & \theta \end{pmatrix}.$$

1. Find the rank of \(\mathbf{A}\) when \(\theta \neq -1\). [3]
2. Find the rank of \(\mathbf{A}\) when \(\theta = -1\). [1]

Consider the system of equations \begin{align} x + 5y + z &= -1,
x - 2y - 2z &= 0,
2x + 3y + \theta z &= \theta. \end{align}

1. Solve the system of equations when \(\theta \neq -1\). [3]
2. Find the general solution when \(\theta = -1\). [3]
3. Show that if \(\theta = -1\) and \(\phi \neq -1\) then \(\mathbf{A}\mathbf{x} = \begin{pmatrix} -1 \\ 0 \\ \phi \end{pmatrix}\) has no solution. [2]

The matrix \(\mathbf{A}\) is defined by $$\mathbf{A} = \begin{pmatrix} 1 & 5 & 1 \\ 1 & -2 & -2 \\ 2 & 3 & \theta \end{pmatrix}.$$

1. Find the rank of \(\mathbf{A}\) when \(\theta \neq -1\). [3]
2. Find the rank of \(\mathbf{A}\) when \(\theta = -1\). [1]

Consider the system of equations \begin{align} x + 5y + z &= -1,
x - 2y - 2z &= 0,
2x + 3y + \theta z &= \theta. \end{align}

1. Solve the system of equations when \(\theta \neq -1\). [3]
2. Find the general solution when \(\theta = -1\). [3]
3. Show that if \(\theta = -1\) and \(\phi \neq -1\) then \(\mathbf{A}\mathbf{x} = \begin{pmatrix} -1 \\ 0 \\ \phi \end{pmatrix}\) has no solution. [2]

Show that the system of equations $$14x - 4y + 6z = 5,$$ $$x + y + kz = 3,$$ $$-21x + 6y - 9z = 14,$$ where \(k\) is a constant, does not have a unique solution and interpret this situation geometrically. [4]

By using the determinant of an appropriate matrix, find the values of \(\lambda\) for which the simultaneous equations \begin{align} 3x + 2y + 4z &= 5,
\lambda y + z &= 1,
x + \lambda y + \lambda z &= 4, \end{align} do not have a unique solution for \(x\), \(y\) and \(z\). [6]

The matrix \(\mathbf{A}\) is given by \(\mathbf{A} = \begin{pmatrix} a & a & -1 \\ 0 & a & 2 \\ 1 & 2 & 1 \end{pmatrix}\).

1. Find, in terms of \(a\), the determinant of \(\mathbf{A}\). [3]
2. Three simultaneous equations are shown below. \begin{align} ax + ay - z &= -1
   ay + 2z &= 2a
   x + 2y + z &= 1 \end{align} For each of the following values of \(a\), determine whether the equations are consistent or inconsistent. If the equations are consistent, determine whether or not there is a unique solution.
   1. \(a = 0\)
   2. \(a = 1\)
   3. \(a = 2\) [6]

Three planes have equations \begin{align} 4x - 5y + z &= 8
3x + 2y - kz &= 6
(k - 2)x + ky - 8z &= 6 \end{align} where \(k\) is a real constant. The planes do not meet at a unique point.

1. Find the possible values of \(k\). [3 marks]
2. For each value of \(k\) found in part (a), identify the configuration of the given planes. Fully justify your answer, stating in each case whether or not the equations of the planes form a consistent system. [5 marks]

Three planes have equations, $$x - y + kz = 3$$ $$kx - 3y + 5z = -1$$ $$x - 2y + 3z = -4$$ Where \(k\) is a real constant. The planes do not meet at a unique point.

1. Find the possible values of \(k\) [3 marks]
2. There are two possible geometric configurations of the given planes. Identify each possible configurations, stating the corresponding value of \(k\) Fully justify your answer. [5 marks]
3. Given further that the equations of the planes form a consistent system, find the solution of the system of equations. [3 marks]

Three planes have equations \begin{align} -x + 2y + z &= 0
2x - y - z &= 0
x + y &= a \end{align} where \(a\) is a constant.

1. Investigate the arrangement of the planes:
   [6]
2. Chris claims that the position vectors \(-\mathbf{i} + 2\mathbf{j} + \mathbf{k}\), \(2\mathbf{i} - \mathbf{j} - \mathbf{k}\) and \(\mathbf{i} + \mathbf{j}\) lie in a plane. Determine whether or not Chris is correct. [2]

$$\mathbf{M} = \begin{pmatrix} 2 & -1 & 1 \\ 3 & k & 4 \\ 3 & 2 & -1 \end{pmatrix} \quad \text{where } k \text{ is a constant}$$

1. Find the values of \(k\) for which the matrix \(\mathbf{M}\) has an inverse. [2]
2. Find, in terms of \(p\), the coordinates of the point where the following planes intersect \begin{align} 2x - y + z &= p
   3x - 6y + 4z &= 1
   3x + 2y - z &= 0 \end{align} [5]
3. 1. Find the value of \(q\) for which the set of simultaneous equations \begin{align} 2x - y + z &= 1
      3x - 5y + 4z &= q
      3x + 2y - z &= 0 \end{align} can be solved.
   2. For this value of \(q\), interpret the solution of the set of simultaneous equations geometrically. [4]

Three planes have equations \begin{align} (4k + 1)x - 3y + (k - 5)z &= 3
(k - 1)x + (3 - k)y + 2z &= 1
7x - 3y + 4z &= 2 \end{align}

1. The planes do not meet at a unique point. Show that \(k = 4.5\) is one possible value of \(k\), and find the other possible value of \(k\). [3 marks]
2. For each value of \(k\) found in part (a), identify the configuration of the given planes. In each case fully justify your answer, stating whether or not the equations of the planes form a consistent system. [4 marks]

Three planes have equations \begin{align} 4x - 5y + z &= 8
3x + 2y - kz &= 6
(k - 2)x + ky - 8z &= 6 \end{align} where \(k\) is a real constant. The planes do not meet at a unique point.

1. Find the possible values of \(k\). [3 marks]
2. For each value of \(k\) found in part (a), identify the configuration of the given planes. Fully justify your answer, stating in each case whether or not the equations of the planes form a consistent system. [5 marks]