4.03s Consistent/inconsistent: systems of equations

2 Find the value of the constant \(k\) for which the system of equations $$\begin{aligned} 2 x - 3 y + 4 z & = 1 \\ 3 x - y & = 2 \\ x + 2 y + k z & = 1 \end{aligned}$$ does not have a unique solution. For this value of \(k\), solve the system of equations.

4

1. Find the value of \(k\) for which the set of linear equations $$\begin{aligned} x + 3 y + k z & = 4 \\ 4 x - 2 y - 10 z & = - 5 \\ x + y + 2 z & = 1 \end{aligned}$$ has no unique solution.
2. For this value of \(k\), find the set of possible solutions, giving your answer in the form $$\left( \begin{array} { c } x \\ y \\ z \end{array} \right) = \mathbf { a } + t \mathbf { b } ,$$ where \(\mathbf { a }\) and \(\mathbf { b }\) are vectors and \(t\) is a scalar.

Let \(V\) be the subspace of \(\mathbb { R } ^ { 4 }\) spanned by $$\mathbf { v } _ { 1 } = \left( \begin{array} { l } 1 \\ 2 \\ 0 \\ 2 \end{array} \right) , \quad \mathbf { v } _ { 2 } = \left( \begin{array} { r } - 2 \\ - 5 \\ 5 \\ 6 \end{array} \right) , \quad \mathbf { v } _ { 3 } = \left( \begin{array} { r } 0 \\ - 3 \\ 15 \\ 18 \end{array} \right) \quad \text { and } \quad \mathbf { v } _ { 4 } = \left( \begin{array} { r } 0 \\ - 2 \\ 10 \\ 8 \end{array} \right) .$$

1. Show that the dimension of \(V\) is 3 . \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)
2. Express \(\mathbf { v } _ { 4 }\) as a linear combination of \(\mathbf { v } _ { 1 } , \mathbf { v } _ { 2 }\) and \(\mathbf { v } _ { 3 }\). \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)
3. Write down a basis for \(V\). \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) Let \(\mathbf { M } = \left( \begin{array} { r r r r } 1 & - 2 & 0 & 0 \\ 2 & - 5 & - 3 & - 2 \\ 0 & 5 & 15 & 10 \\ 2 & 6 & 18 & 8 \end{array} \right)\).
4. Find the general solution of \(\mathbf { M x } = \mathbf { v } _ { 1 } + \mathbf { v } _ { 2 }\). \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) The set of elements of \(\mathbb { R } ^ { 4 }\) which are not solutions of \(\mathbf { M x } = \mathbf { v } _ { 1 } + \mathbf { v } _ { 2 }\) is denoted by \(W\).
5. State, with a reason, whether \(W\) is a vector space. \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)

The linear transformation \(\mathrm { T } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }\) is represented by the matrix $$\mathbf { M } = \left( \begin{array} { r r r r } - 1 & 2 & 3 & 4 \\ 1 & 0 & 1 & - 1 \\ 1 & - 2 & - 3 & a \\ 1 & 2 & 5 & 2 \end{array} \right) .$$

1. For \(a \neq - 4\), the range space of T is denoted by \(V\).
   1. Find the dimension of \(V\) and show that $$\left( \begin{array} { r } - 1 \\ 1 \\ 1 \\ 1 \end{array} \right) , \quad \left( \begin{array} { r } 2 \\ 0 \\ - 2 \\ 2 \end{array} \right) \quad \text { and } \quad \left( \begin{array} { r } 4 \\ - 1 \\ a \\ 2 \end{array} \right)$$ form a basis for \(V\).
   2. Show that if \(\left( \begin{array} { l } x \\ y \\ z \\ t \end{array} \right)\) belongs to \(V\) then \(x + 2 y = t\).
   3. For \(a = - 4\), find the general solution of $$\mathbf { M } \mathbf { x } = \left( \begin{array} { r } - 1 \\ 1 \\ 1 \\ 1 \end{array} \right)$$ If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

5 The linear transformation \(\mathrm { T } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }\) is represented by the matrix \(\mathbf { M }\), where $$\mathbf { M } = \left( \begin{array} { r r r r } 1 & 2 & 0 & 4 \\ 5 & 2 & 1 & - 3 \\ 4 & 0 & 1 & - 7 \\ - 2 & 4 & - 1 & \alpha \end{array} \right)$$ It is given that the rank of \(\mathbf { M }\) is 2 .

1. Find the value of \(\alpha\) and state a basis for the range space of T .
2. Obtain a basis for the null space of T .

9 The planes \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\), which meet in the line \(/\), have vector equations $$\begin{aligned} & \mathbf { r } = 2 \mathbf { i } + 4 \mathbf { j } + 6 \mathbf { k } + \theta _ { 1 } ( 2 \mathbf { i } + 3 \mathbf { k } ) + \phi _ { 1 } ( - 4 \mathbf { j } + 5 \mathbf { k } ) , \\ & \mathbf { r } = 2 \mathbf { i } + 4 \mathbf { j } + 6 \mathbf { k } + \theta _ { 2 } ( 3 \mathbf { j } + \mathbf { k } ) + \phi _ { 2 } ( - \mathbf { i } + \mathbf { j } + 2 \mathbf { k } ) , \end{aligned}$$ respectively. Find a vector equation of the line \(l\) in the form \(\mathbf { r } = \mathbf { a } + t \mathbf { b }\). Find a vector equation of the plane \(\Pi _ { 3 }\) which contains \(l\) and which passes through the point with position vector \(4 \mathbf { i } + 3 \mathbf { j } + 2 \mathbf { k }\). Find also the equation of \(\Pi _ { 3 }\) in the form \(a x + b y + c z = d\). Deduce, or prove otherwise, that the system of equations $$\begin{aligned} & 6 x - 5 y - 4 z = - 32 \\ & 5 x - y + 3 z = 24 \\ & 9 x - 2 y + 5 z = 40 \end{aligned}$$ has an infinite number of solutions.

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.

6 The matrix \(\mathbf { A }\) is defined by $$\mathbf { A } = \left( \begin{array} { r r r r } 1 & - 1 & - 2 & - 3 \\ - 2 & 1 & 7 & 2 \\ - 3 & 3 & 6 & \alpha \\ 7 & - 6 & - 17 & - 17 \end{array} \right) .$$

1. Show that if \(\alpha = 9\) then the rank of \(\mathbf { A }\) is 2, and find a basis for the null space of \(\mathbf { A }\) in this case.
2. Find the rank of \(\mathbf { A }\) when \(\alpha \neq 9\).

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 \(b \neq - 5\).

10 The matrix \(\mathbf { A }\) is defined by $$\mathbf { A } = \left( \begin{array} { r r r } 1 & 5 & 1 \\ 1 & - 2 & - 2 \\ 2 & 3 & \theta \end{array} \right)$$

1. (a) Find the rank of \(\mathbf { A }\) when \(\theta \neq - 1\).
   (b) Find the rank of \(\mathbf { A }\) when \(\theta = - 1\).
   Consider the system of equations $$\begin{aligned} x + 5 y + z & = - 1 \\ x - 2 y - 2 z & = 0 \\ 2 x + 3 y + \theta z & = \theta \end{aligned}$$
2. Solve the system of equations when \(\theta \neq - 1\).
3. Find the general solution when \(\theta = - 1\).
4. Show that if \(\theta = - 1\) and \(\phi \neq - 1\) then \(\mathbf { A } \mathbf { x } = \left( \begin{array} { r } - 1 \\ 0 \\ \phi \end{array} \right)\) has no solution.

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 $$\mathbf { M x } = \left( \begin{array} { l } 8 \\ 7 \\ k \end{array} \right)$$

2 Find the set of values of \(a\) for which the system of equations $$\begin{aligned} a x + y + 2 z & = 0 \\ 3 x - 2 y & = 4 \\ 3 x - 4 y - 6 a z & = 14 \end{aligned}$$ has a unique solution.

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 \(K\).

10 You are given the matrix \(\mathbf { A }\) where \(\mathbf { A } = \left( \begin{array} { l l l } a & 2 & 0 \\ 0 & a & 2 \\ 4 & 5 & 1 \end{array} \right)\).

1. Find, in terms of \(a\), the determinant of \(\mathbf { A }\), simplifying your answer.
2. Hence find the values of \(a\) for which \(\mathbf { A }\) is singular. You are given the following equations which are to be solved simultaneously. $$\begin{aligned} a x + 2 y & = 6 \\ a y + 2 z & = 8 \\ 4 x + 5 y + z & = 16 \end{aligned}$$
3. For each of the values of \(a\) found in part (b) determine whether the equations have

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

7 You are given that \(a\) is a parameter which can take only real values.
The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { c c r } 2 & 4 & - 6 \\ - 3 & 10 - 4 a & 9 \\ 7 & 4 & 4 \end{array} \right)\).

1. Find an expression for the determinant of \(\mathbf { A }\) in terms of \(a\). You are given the following system of equations in \(x , y\) and \(z\). $$\begin{array} { r r } 2 x + & 4 y - 6 z = \\ - 3 x + & ( 10 - 4 a ) y + 9 z = \\ 7 x + & 4 y + 4 z = \\ 7 x + & 11 \end{array}$$ The system can be written in the form \(\mathbf { A } \left( \begin{array} { c } \mathrm { x } \\ \mathrm { y } \\ \mathrm { z } \end{array} \right) = \left( \begin{array} { r } 6 \\ - 9 \\ 11 \end{array} \right)\).
2. 1. In the case where \(\mathbf { A }\) is not singular, solve the given system of equations by using \(\mathbf { A } ^ { - 1 }\).
   2. In the case where \(\mathbf { A }\) is singular describe the configuration of the planes whose equations are the three equations of the system. The transformation represented by \(\mathbf { A }\) is denoted by T .
      A 3-D object of volume \(| 5 a - 20 |\) is transformed by T to a 3-D image.
3. 1. Determine the range of values of \(a\) for which the orientation of the image is the reverse of the orientation of the object.
   2. Determine the range of values of \(a\) for which the volume of the image is less than the volume of the object.

8 The equations of three planes are $$\begin{array} { r } 2 x + y + 3 z = 3 \\ 3 x - y - 2 z = 2 \\ - 4 x + 3 y + 7 z = k \end{array}$$ where \(k\) is a constant.

1. By considering a suitable determinant, show that the planes do not meet at a single point.
2. Given that the planes form a sheaf, determine the value of \(k\).

7 Three planes have equations $$\begin{array} { r } x + 2 y - 3 z = 0 \\ - x + 3 y - 2 z = 0 \\ x - 2 y + k z = k \end{array}$$ where \(k\) is a constant.

1. For the case \(k = 0\), the origin lies on all three planes. Use a determinant to explain whether there are any other points that lie on all three planes in this case.
2. You are now given that \(k = 1\).
   1. Show that there are no points that lie on all three planes.
   2. Describe the geometrical arrangement of the three planes.

9 Three planes have equations \(k x + y - 2 z = 0\) \(2 x + 3 y - 6 z = - 5\) \(3 x - 2 y + 5 z = 1\) where \(k\) is a constant. Investigate the arrangement of the planes for each of the following cases. If in either case the planes meet at a unique point, find the coordinates of that point.

1. \(k = - 1\)
2. \(k = \frac { 2 } { 3 }\)

14 Three planes have equations $$\begin{aligned} - x + a y & = 2 \\ 2 x + 3 y + z & = - 3 \\ x + b y + z & = c \end{aligned}$$ where \(a\), \(b\) and \(c\) are constants.

1. In the case where the planes do not intersect at a unique point,
   1. find \(b\) in terms of \(a\),
   2. find the value of \(c\) for which the planes form a sheaf.
2. In the case where \(b = a\) and \(c = 1\), find the coordinates of the point of intersection of the planes in terms of \(a\).

14 Three planes have equations $$\begin{aligned} k x - z & = 2 \\ - x + k y + 2 z & = 1 \\ 2 k x + 2 y + 3 z & = 0 \end{aligned}$$ where \(k\) is a constant.

1. By considering a suitable determinant, show that the three planes meet at a point for all values of \(k\).
2. Using a matrix method, find, in terms of \(k\), the coordinates of the point of intersection of the planes.

15 Three planes have equations $$\begin{aligned} x + k y + 3 z & = 1 \\ 3 x + 4 y + 2 z & = 3 \\ x + 3 y - z & = - k \end{aligned}$$ where \(k\) is a constant.

1. Show that the planes meet at a point except for one value of \(k\), which should be determined.
2. Show that, when the planes do meet at a point, the \(y\)-coordinate of this point is independent of \(k\).

15

1. Show that the three planes with equations $$\begin{aligned} x + \lambda y + 3 z & = - 12 \\ 2 x + y + 5 z & = - 11 \\ x - 2 y + 2 z & = - 9 \end{aligned}$$ where \(\lambda\) is a constant, meet at a unique point except for one value of \(\lambda\) which is to be determined.
2. In the case \(\lambda = - 2\), use matrices to find the point of intersection P of the planes, showing your method clearly. The line \(l\) has equation \(\frac { x - 1 } { 2 } = \frac { y - 1 } { - 1 } = \frac { z + 2 } { - 2 }\).
3. Find a vector equation of \(l\).
4. Find the shortest distance between the point P and \(l\).
5. 1. Show that \(l\) is parallel to the plane \(x - 2 y + 2 z = - 9\).
   2. Find the distance between \(l\) and the plane \(x - 2 y + 2 z = - 9\).