4.03r Solve simultaneous equations: using inverse matrix

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 .

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.

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

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 }

2 Show that the matrix \(\left( \begin{array} { r r r } 1 & 4 & 2 \\ 3 & 0 & - 2 \\ 3 & - 3 & - 4 \end{array} \right)\) has no inverse. Solve the system of equations $$\begin{array} { r } x + 4 y + 2 z = 0 \\ 3 x - 2 z = 4 \\ 3 x - 3 y - 4 z = 5 \end{array}$$

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.

7 The linear transformation \(\mathrm { T } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }\) is represented by the matrix \(\mathbf { M }\), where $$\mathbf { M } = \left( \begin{array} { r r r r } 1 & - 2 & - 3 & 1 \\ 3 & - 5 & - 7 & 7 \\ 5 & - 9 & - 13 & 9 \\ 7 & - 13 & - 19 & 11 \end{array} \right)$$

1. Find the rank of \(\mathbf { M }\) and a basis for the null space of T .
2. The vector \(\left( \begin{array} { l } 1 \\ 2 \\ 3 \\ 4 \end{array} \right)\) is denoted by \(\mathbf { e }\). Show that there is a solution of the equation \(\mathbf { M x } = \mathbf { M e }\) of the form \(\mathbf { x } = \left( \begin{array} { c } a \\ b \\ - 1 \\ - 1 \end{array} \right)\), where the constants \(a\) and \(b\) are to be found.

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

1. Determine the dimension of \(V\).
2. Show that the vectors \(\left( \begin{array} { l } 1 \\ 1 \\ 1 \\ 1 \end{array} \right) , \left( \begin{array} { r } 2 \\ - 1 \\ - 3 \\ 4 \end{array} \right) , \left( \begin{array} { l } 3 \\ 2 \\ 3 \\ 2 \end{array} \right)\) are a basis of \(V\). The set of elements of \(\mathbb { R } ^ { 4 }\) which do not belong to \(V\) is denoted by \(W\).
3. State, with a reason, whether \(W\) is a vector space.
4. Show that if the vector \(\left( \begin{array} { l } x \\ y \\ z \\ t \end{array} \right)\) belongs to \(W\) then \(x + y \neq z + t\).

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 Use a matrix method to determine the solution of the following simultaneous equations. $$\begin{aligned} 2 x - 3 y + z & = 1 \\ x - 2 y - 4 z & = 40 \\ 5 x + 6 y - z & = 61 \end{aligned}$$

6 At the beginning of the year John had a total of \(\pounds 2000\) in three different accounts. He has twice as much money in the current account as in the savings account.

• The current account has an interest rate of \(2.5 \%\) per annum.
• The savings account has an interest rate of \(3.7 \%\) per annum.
• The supersaver account has an interest rate of \(4.9 \%\) per annum.

John has predicted that he will earn a total interest of \(\pounds 92\) by the end of the year.

1. Model this situation as a matrix equation.
2. Find the amount that John had in each account at the beginning of the year.
3. In fact, the interest John will receive is \(\pounds 92\) to the nearest pound. Explain how this affects the calculations.

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

2 The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { r r } 2 & - 2 \\ 1 & 3 \end{array} \right)\).

1. Calculate \(\operatorname { det } \mathbf { A }\).
2. Write down \(\mathbf { A } ^ { - 1 }\).
3. Hence solve the equation \(\mathbf { A } \binom { \mathrm { x } } { \mathrm { y } } = \binom { - 1 } { 2 }\).
4. Write down the matrix \(\mathbf { B }\) such that \(\mathbf { A B } = 4 \mathbf { I }\). Matrices \(\mathbf { C }\) and \(\mathbf { D }\) are given by \(\mathbf { C } = \left( \begin{array} { l } 2 \\ 0 \\ 1 \end{array} \right)\) and \(\mathbf { D } = \left( \begin{array} { l l l } 0 & 2 & p \end{array} \right)\) where \(p\) is a constant.
5. Find, in terms of \(p\),
   It is observed that \(\mathbf { C D } \neq \mathbf { D C }\).
6. The result that \(\mathbf { C D } \neq \mathbf { D C }\) is a counter example to the claim that matrix multiplication has a particular property. Name this property.

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

1

1. 1. Write the following simultaneous equations as a matrix equation. $$\begin{aligned} x + y + 2 z & = 7 \\ 2 x - 4 y - 3 z & = - 5 \\ - 5 x + 3 y + 5 z & = 13 \end{aligned}$$
   2. Hence solve the equations.
2. Determine the set of values of the constant \(k\) for which the matrix equation $$\left( \begin{array} { c c } k + 1 & 1 \\ 2 & k \end{array} \right) \binom { x } { y } = \binom { 23 } { - 17 }$$ has a unique solution.

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

3 Three planes have the following equations. $$\begin{aligned} 2 x - 3 y + z & = - 3 \\ x - 4 y + 2 z & = 1 \\ - 3 x - 2 y + 3 z & = 14 \end{aligned}$$

1. 1. Write the system of equations in matrix form.
   2. Hence find the point of intersection of the planes.
2. In this question you must show detailed reasoning. Find the acute angle between the planes \(2 x - 3 y + z = - 3\) and \(x - 4 y + 2 z = 1\).

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 \(x - 2 y + 2 z = - 9\).

1. The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by \(\mathbf { A } = \left( \begin{array} { r r } 3 & 7 \\ - 2 & 0 \end{array} \right)\), \(\mathbf { B } = \left( \begin{array} { l l } 5 & 1 \\ 0 & 4 \end{array} \right)\).

The matrix \(\mathbf { X }\) is such that \(\mathbf { A X } = \mathbf { B }\). Showing all your working, find the matrix \(\mathbf { X }\).

2. (a) The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are defined by $$\mathbf { A } = \left( \begin{array} { c c } 3 & 4 \\ - 1 & - 2 \end{array} \right) , \quad \mathbf { B } = \binom { - 11 } { 7 }$$ Given that \(\mathbf { A X } = \mathbf { B }\), find the matrix \(\mathbf { X }\).
(b) (i) Find the \(2 \times 2\) matrix, \(\mathbf { T }\), which represents a reflection in the line \(y = - 2 x\).
(ii) The images of the points \(C ( 2,7 )\) and \(D ( 3,1 )\), under \(\mathbf { T }\), are \(E\) and \(F\) respectively. Find the coordinates of the midpoint of \(E F\).