4.03t Plane intersection: geometric interpretation

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).

6 The matrix \(\mathbf { A }\) is given by $$A = \left( \begin{array} { r r r } 5 & - \frac { 22 } { 3 } & 8 \\ 0 & - 6 & 0 \\ 0 & 0 & 1 \end{array} \right)$$

1. Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { A } ^ { 2 } = \mathbf { P D P } ^ { - 1 }\).
2. Use the characteristic equation of \(\mathbf { A }\) to find \(\mathbf { A } ^ { 3 }\).

8

1. Find the value of \(a\) for which the system of equations $$\begin{array} { r } 13 x + 18 y - 28 z = 0 \\ - 4 x - a y + 8 z = 0 \\ 2 x + 6 y - 5 z = 0 \end{array}$$ does not have a unique solution.
   The matrix \(\mathbf { A }\) is given by $$\mathbf { A } = \left( \begin{array} { r r r } 13 & 18 & - 28 \\ - 4 & - 1 & 8 \\ 2 & 6 & - 5 \end{array} \right)$$
2. Find the eigenvalue of \(\mathbf { A }\) corresponding to the eigenvector \(\left( \begin{array} { l } 2 \\ 0 \\ 1 \end{array} \right)\).
3. Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { A } = \mathbf { P D P } ^ { - 1 }\).
4. Use the characteristic equation of \(\mathbf { A }\) to find \(\mathbf { A } ^ { - 1 }\) in terms of \(\mathbf { A }\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

8 The matrix \(\mathbf { A }\) is given by $$\mathbf { A } = \left( \begin{array} { c c c } a & - 6 a & 2 a + 2 \\ 0 & 1 - a & 0 \\ 0 & 2 - a & - 1 \end{array} \right)$$ where \(a\) is a constant with \(a \neq 0\) and \(a \neq 1\).

1. Show that the equation \(\mathbf { A } \left( \begin{array} { c } x \\ y \\ z \end{array} \right) = \left( \begin{array} { c } 1 \\ 2 \\ 3 \end{array} \right)\) has a unique solution and interpret this situation geometrically.
2. Show that the eigenvalues of \(\mathbf { A }\) are \(a , 1 - a\) and - 1 .
3. Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { A } ^ { 4 } = \mathbf { P D P } ^ { - 1 }\).
4. Use the characteristic equation of \(\mathbf { A }\) to find \(\mathbf { A } ^ { 4 }\) in terms of \(\mathbf { A }\) and \(a\).
   If you use the following page to complete the answer to any question, the question number must be clearly shown.

8

1. Find the set of values of \(a\) for which the system of equations $$\begin{array} { c l } 6 x + a y & = 3 \\ 2 x - y & = 1 \\ x + 5 y + 4 z & = 2 \end{array}$$ has a unique solution.
2. Show that the system of equations in part (a) is consistent for all values of \(a\).
   The matrix \(\mathbf { A }\) is given by $$\mathbf { A } = \left( \begin{array} { r r r } 6 & 0 & 0 \\ 2 & - 1 & 0 \\ 1 & 5 & 4 \end{array} \right)$$
3. Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(( 14 \mathbf { A } + 24 \mathbf { I } ) ^ { 2 } = \mathbf { P D P } ^ { - 1 }\).
4. Use the characteristic equation of \(\mathbf { A }\) to show that $$( 14 \mathbf { A } + 24 \mathbf { I } ) ^ { 2 } = \mathbf { A } ^ { 4 } ( \mathbf { A } + b \mathbf { I } ) ^ { 2 }$$ where \(b\) is an integer to be determined.
   If you use the following page to complete the answer to any question, the question number must be clearly shown.

3

1. Show that the system of equations $$\begin{array} { r } x - 2 y - 4 z = 1 \\ x - 2 y + k z = 1 \\ - x + 2 y + 2 z = 1 \end{array}$$ where \(k\) is a constant, does not have a unique solution.
2. Given that \(k = - 4\), show that the system of equations in part (a) is consistent. Interpret this situation geometrically.
3. Given instead that \(k = - 2\), show that the system of equations in part (a) is inconsistent. Interpret this situation geometrically.
4. For the case where \(k \neq - 2\) and \(k \neq - 4\), show that the system of equations in part (a) is inconsistent. Interpret this situation geometrically. \includegraphics[max width=\textwidth, alt={}, center]{7da7fa35-1b97-4708-a1a2-cba9e35c8bf0-06_894_841_260_612} The diagram shows the curve with equation \(\mathrm { y } = 1 - \mathrm { x } ^ { 3 }\) for \(0 \leqslant x \leqslant 1\), together with a set of \(n\) rectangles of width \(\frac { 1 } { n }\).

3

1. Show that the system of equations $$\begin{array} { r } x - 2 y - 4 z = 1 \\ x - 2 y + k z = 1 \\ - x + 2 y + 2 z = 1 \end{array}$$ where \(k\) is a constant, does not have a unique solution.
2. Given that \(k = - 4\), show that the system of equations in part (a) is consistent. Interpret this situation geometrically.
3. Given instead that \(k = - 2\), show that the system of equations in part (a) is inconsistent. Interpret this situation geometrically.
4. For the case where \(k \neq - 2\) and \(k \neq - 4\), show that the system of equations in part (a) is inconsistent. Interpret this situation geometrically. \includegraphics[max width=\textwidth, alt={}, center]{23c7189f-850d-4745-a8ce-46a140ed0176-06_894_841_260_612} The diagram shows the curve with equation \(\mathrm { y } = 1 - \mathrm { x } ^ { 3 }\) for \(0 \leqslant x \leqslant 1\), together with a set of \(n\) rectangles of width \(\frac { 1 } { n }\).

1. The plane \(\Pi _ { 1 }\) has vector equation

$$\mathbf { r } = \left( \begin{array} { l } 5 \\ 3 \\ 0 \end{array} \right) + s \left( \begin{array} { l } 3 \\ 0 \\ 1 \end{array} \right) + t \left( \begin{array} { r } 1 \\ - 2 \\ 2 \end{array} \right)$$ where \(s\) and \(t\) are scalar parameters.

1. Determine a Cartesian equation for \(\Pi _ { 1 }\) The plane \(\Pi _ { 2 }\) has vector equation \(\mathbf { r } . \left( \begin{array} { r } 5 \\ - 2 \\ 3 \end{array} \right) = 1\)
2. Determine a vector equation for the line of intersection of \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\) Give your answer in the form \(\mathbf { r } = \mathbf { a } + \lambda \mathbf { b }\), where \(\mathbf { a }\) and \(\mathbf { b }\) are constant vectors and \(\lambda\) is a scalar parameter. The plane \(\Pi _ { 3 }\) has Cartesian equation \(4 x - 3 y - z = 0\)
3. Use the answer to part (b) to determine the coordinates of the point of intersection of \(\Pi _ { 1 } , \Pi _ { 2 }\) and \(\Pi _ { 3 }\)

9 The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { l l l } a & 1 & 1 \\ 1 & a & 1 \\ 1 & 1 & 2 \end{array} \right)\).

1. Find, in terms of \(a\), the determinant of \(\mathbf { A }\).
2. Hence find the values of \(a\) for which \(\mathbf { A }\) is singular.
3. State, giving a brief reason in each case, whether the simultaneous equations $$\begin{aligned} a x + y + z & = 2 a \\ x + a y + z & = - 1 \\ x + y + 2 z & = - 1 \end{aligned}$$ have any solutions when
   1. \(a = 0\),
   2. \(a = 1\).

The linear transformation \(\mathrm { T } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 3 }\) is represented by the matrix \(\mathbf { M }\), where $$\mathbf { M } = \left( \begin{array} { r r r r } 2 & 1 & - 1 & 4 \\ 3 & 4 & 6 & 1 \\ - 1 & 2 & 8 & - 7 \end{array} \right)$$ The range space of T is \(R\). In any order,

1. show that the dimension of \(R\) is 2 ,
2. find a basis for \(R\) and obtain a cartesian equation for \(R\),
3. find a basis for the null space of T . The vector \(\left( \begin{array} { l } 8 \\ 7 \\ k \end{array} \right)\) belongs to \(R\). Find the value of \(k\) and, with this value of \(k\), find the general solution of $$\mathbf { M } \mathbf { x } = \left( \begin{array} { l } 8 \\ 7 \\ k \end{array} \right) .$$

5 Find the value of \(a\) for which the system of equations $$\begin{aligned} & x - y + 2 z = 4 \\ & x + a y - 3 z = b \\ & x - y + 7 z = 13 \end{aligned}$$ where \(a\) and \(b\) are constants, has no unique solution. Taking \(a\) as the value just found,

1. find the general solution in the case \(b = - 5\),
2. interpret the situation geometrically in the case \(b \neq - 5\).

4 The linear transformation \(\mathrm { T } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }\) is represented by the matrix \(\mathbf { M }\), where $$\mathbf { M } = \left( \begin{array} { r r r r } 3 & 4 & 2 & 5 \\ 6 & 7 & 5 & 8 \\ 9 & 9 & 9 & 9 \\ 15 & 16 & 14 & 17 \end{array} \right)$$ Find

1. the rank of \(\mathbf { M }\) and a basis for the range space of T ,
2. a basis for the null space of T .

The linear transformation \(\mathrm { T } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 3 }\) is represented by the matrix \(\mathbf { M }\), where $$\mathbf { M } = \left( \begin{array} { r r r r } 2 & 1 & - 1 & 4 \\ 3 & 4 & 6 & 1 \\ - 1 & 2 & 8 & - 7 \end{array} \right)$$ The range space of T is \(R\). In any order,

1. show that the dimension of \(R\) is 2 ,
2. find a basis for \(R\) and obtain a cartesian equation for \(R\),
3. find a basis for the null space of T . The vector \(\left( \begin{array} { l } 8 \\ 7 \\ k \end{array} \right)\) belongs to \(R\). Find the value of \(k\) and, with this value of \(k\), find the general solution of $$\mathbf { M x } = \left( \begin{array} { l } 8 \\ 7 \\ k \end{array} \right)$$

6 The linear transformation \(\mathrm { T } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }\) is represented by the matrix \(\mathbf { M }\), where $$\mathbf { M } = \left( \begin{array} { r r r r } 1 & - 3 & - 1 & 2 \\ 4 & - 10 & 0 & 2 \\ 1 & - 1 & 3 & - 4 \\ 5 & - 12 & 1 & 1 \end{array} \right)$$ Find, in either order, the rank of \(\mathbf { M }\) and a basis for the null space \(K\) of T . Evaluate $$\mathbf { M } \left( \begin{array} { r } 1 \\ - 2 \\ - 3 \\ - 4 \end{array} \right)$$ and hence show that every solution of $$\mathbf { M x } = \left( \begin{array} { r } 2 \\ 16 \\ 10 \\ 22 \end{array} \right)$$ has the form $$\mathbf { x } = \left( \begin{array} { r } 1 \\ - 2 \\ - 3 \\ - 4 \end{array} \right) + \lambda \mathbf { e } _ { 1 } + \mu \mathbf { e } _ { 2 } ,$$ where \(\lambda\) and \(\mu\) are real numbers and \(\left\{ \mathbf { e } _ { 1 } , \mathbf { e } _ { 2 } \right\}\) is a basis for \(K\).

1. $$\mathbf { M } = \left( \begin{array} { r r r } 2 & 1 & - 3 \\ 4 & - 2 & 1 \\ 3 & 5 & - 2 \end{array} \right)$$

1. Find \(\mathbf { M } ^ { - 1 }\) giving each element in exact form.
2. Solve the simultaneous equations $$\begin{array} { r } 2 x + y - 3 z = - 4 \\ 4 x - 2 y + z = 9 \\ 3 x + 5 y - 2 z = 5 \end{array}$$
3. Interpret the answer to part (b) geometrically.

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4. $$\mathbf { M } = \left( \begin{array} { r r r } 2 & 1 & 4 \\ k & 2 & - 2 \\ 4 & 1 & - 2 \end{array} \right) \quad \mathbf { N } = \left( \begin{array} { r r r } k - 7 & 6 & - 10 \\ 2 & - 20 & 24 \\ - 3 & 2 & - 1 \end{array} \right)$$ where \(k\) is a constant.

1. Determine, in simplest form in terms of \(k\), the matrix \(\mathbf { M N }\).
2. Given that \(k = 5\)
   1. write down \(\mathbf { M N }\)
   2. hence write down \(\mathbf { M } ^ { - 1 }\)
3. Solve the simultaneous equations $$\begin{aligned} & 2 x + y + 4 z = 2 \\ & 5 x + 2 y - 2 z = 3 \\ & 4 x + y - 2 z = - 1 \end{aligned}$$
4. Interpret the answer to part (c) 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.

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.

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]

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]