4.03r Solve simultaneous equations: using inverse matrix

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).

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

1. Find the value of \(a\) for which the system of equations $$\begin{gathered} 3 x + a y = 0 \\ 5 x - y = 0 \\ x + 3 y + 2 z = 0 \end{gathered}$$ does not have a unique solution.
   The matrix \(\mathbf { A }\) is given by $$\mathbf { A } = \left( \begin{array} { r r r } 3 & 0 & 0 \\ 5 & - 1 & 0 \\ 1 & 3 & 2 \end{array} \right)$$
2. Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { A } ^ { 2 } = \mathbf { P D P } ^ { - 1 }\).
3. Use the characteristic equation of \(\mathbf { A }\) to show that $$( \mathbf { A } + 6 \mathbf { I } ) ^ { 2 } = \mathbf { A } ^ { 4 } ( \mathbf { A } + b \mathbf { I } ) ^ { 2 }$$ 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.

1

1. Show that the system of equations $$\begin{array} { r } x + 2 y + 3 z = 1 \\ 4 x + 5 y + 6 z = 1 \\ 7 x + 8 y + 9 z = 1 \end{array}$$ does not have a unique solution.
2. Show that the system of equations in part (a) is consistent. Interpret this situation geometrically.

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.

2

1. Show that the system of equations $$\begin{aligned} & x - y + 2 z = 4 \\ & x - y - 3 z = a \\ & x - y + 7 z = 13 \end{aligned}$$ where \(a\) is a constant, does not have a unique solution.
2. Given that \(a = - 5\), show that the system of equations in part (a) is consistent. Interpret this situation geometrically.
3. Given instead that \(a \neq - 5\), show that the system of equations in part (a) is inconsistent. Interpret this situation geometrically.

8

1. 1. Find the set of values of \(a\) for which the system of equations $$\begin{array} { r } x - 2 y - 2 z + 7 = 0

7. $$\mathbf { P } = \left( \begin{array} { c c } 3 a & - 2 a \\ - b & 2 b \end{array} \right) , \quad \mathbf { M } = \left( \begin{array} { c c } - 6 a & 7 a \\ 2 b & - b \end{array} \right)$$ where \(a\) and \(b\) are non-zero constants.

1. Find \(\mathbf { P } ^ { - 1 }\), leaving your answer in terms of \(a\) and \(b\). Given that $$\mathbf { M } = \mathbf { P Q }$$
2. find the matrix \(\mathbf { Q }\), giving your answer in its simplest form. \includegraphics[max width=\textwidth, alt={}, center]{9093bb1d-4f32-44e7-b0e7-b8c4f8a844e1-19_95_77_2617_1804}

5. \(\mathbf { A } = \left( \begin{array} { r r } - 4 & a \\ b & - 2 \end{array} \right)\), where \(a\) and \(b\) are constants. Given that the matrix \(\mathbf { A }\) maps the point with coordinates \(( 4,6 )\) onto the point with coordinates \(( 2 , - 8 )\),

1. find the value of \(a\) and the value of \(b\). A quadrilateral \(R\) has area 30 square units.
   It is transformed into another quadrilateral \(S\) by the matrix \(\mathbf { A }\).
   Using your values of \(a\) and \(b\),
2. find the area of quadrilateral \(S\).

4. A non-singular matrix \(\mathbf { M }\) is given by $$\mathbf { M } = \left( \begin{array} { l l l } 3 & k & 0 \\ k & 2 & 0 \\ k & 0 & 1 \end{array} \right) \text {, where } k \text { is a constant. }$$

1. Find, in terms of \(k\), the inverse of the matrix \(\mathbf { M }\). The point \(A\) is mapped onto the point ( \(- 5,10,7\) ) by the transformation represented by the matrix $$\left( \begin{array} { l l l } 3 & 1 & 0 \\ 1 & 2 & 0 \\ 1 & 0 & 1 \end{array} \right)$$
2. Find the coordinates of the point \(A\).

4. $$\mathbf { M } = \left( \begin{array} { r r r } 1 & k & 0 \\ - 1 & 1 & 1 \\ 1 & k & 3 \end{array} \right) , \text { where } k \text { is a constant }$$

1. Find \(\mathbf { M } ^ { - 1 }\) in terms of \(k\). Hence, given that \(k = 0\)
2. find the matrix \(\mathbf { N }\) such that $$\mathbf { M N } = \left( \begin{array} { r r r } 3 & 5 & 6 \\ 4 & - 1 & 1 \\ 3 & 2 & - 3 \end{array} \right)$$

3

1. Find the inverse of the matrix \(\left( \begin{array} { r r r } 4 & 1 & k \\ 3 & 2 & 5 \\ 8 & 5 & 13 \end{array} \right)\), where \(k \neq 5\).
2. Solve the simultaneous equations $$\begin{aligned} & 4 x + y + 7 z = 12 \\ & 3 x + 2 y + 5 z = m \\ & 8 x + 5 y + 13 z = 0 \end{aligned}$$ giving \(x , y\) and \(z\) in terms of \(m\).
3. Find the value of \(p\) for which the simultaneous equations $$\begin{aligned} & 4 x + y + 5 z = 12 \\ & 3 x + 2 y + 5 z = p \\ & 8 x + 5 y + 13 z = 0 \end{aligned}$$ have solutions, and find the general solution in this case.

10 The matrix \(\mathbf { D }\) is given by \(\mathbf { D } = \left( \begin{array} { r r r } a & 2 & 0 \\ 3 & 1 & 2 \\ 0 & - 1 & 1 \end{array} \right)\), where \(a \neq 2\).

1. Find \(\mathbf { D } ^ { - 1 }\).
2. Hence, or otherwise, solve the equations $$\begin{aligned} a x + 2 y & = 3 \\ 3 x + y + 2 z & = 4 \\ - y + z & = 1 \end{aligned}$$

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

1. Given that \(\mathbf { A }\) is singular, find \(a\).
2. Given instead that \(\mathbf { A }\) is non-singular, find \(\mathbf { A } ^ { - 1 }\) and hence solve the simultaneous equations $$\begin{aligned} a x + 3 y & = 1 \\ - 2 x + y & = - 1 \end{aligned}$$

8 The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { l l l } a & 4 & 2 \\ 1 & a & 0 \\ 1 & 2 & 1 \end{array} \right)\).

1. Find, in terms of \(a\), the determinant of \(\mathbf { M }\).
2. Hence find the values of \(a\) for which \(\mathbf { M }\) is singular.
3. State, giving a brief reason in each case, whether the simultaneous equations $$\begin{aligned} a x + 4 y + 2 z & = 3 a \\ x + a y & = 1 \\ x + 2 y + z & = 3 \end{aligned}$$ have any solutions when
   1. \(a = 3\),
   2. \(a = 2\).

7 The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { l l l } a & 4 & 0 \\ 0 & a & 4 \\ 2 & 3 & 1 \end{array} \right)\).

1. Find, in terms of \(a\), the determinant of \(\mathbf { M }\).
2. In the case when \(a = 2\), state whether \(\mathbf { M }\) is singular or non-singular, justifying your answer.
3. In the case when \(a = 4\), determine whether the simultaneous equations $$\begin{aligned} a x + 4 y \quad = & 6 \\ a y + 4 z & = 8 \\ 2 x + 3 y + z & = 1 \end{aligned}$$ have any solutions.

10 The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { r r r } a & 8 & 10 \\ 2 & 1 & 2 \\ 4 & 3 & 6 \end{array} \right)\). The matrix \(\mathbf { B }\) is such that \(\mathbf { A B } = \left( \begin{array} { l l l } a & 6 & 1 \\ 1 & 1 & 0 \\ 1 & 3 & 0 \end{array} \right)\).

1. Show that \(\mathbf { A B }\) is non-singular.
2. Find \(( \mathbf { A B } ) ^ { - 1 }\).
3. Find \(\mathbf { B } ^ { - 1 }\).

10 The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { l l l } a & 2 & 1 \\ 1 & 3 & 2 \\ 4 & 1 & 1 \end{array} \right)\).

1. Find the value of \(a\) for which \(\mathbf { A }\) is singular.
2. Given that \(\mathbf { A }\) is non-singular, find \(\mathbf { A } ^ { - 1 }\) and hence solve the equations $$\begin{aligned} a x + 2 y + z & = 1 \\ x + 3 y + 2 z & = 2 \\ 4 x + y + z & = 3 \end{aligned}$$

8 The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { r r r } a & 2 & - 1 \\ 2 & 3 & - 1 \\ 2 & - 1 & 1 \end{array} \right)\), where \(a\) is a constant.

1. Show that the determinant of \(\mathbf { M }\) is \(2 a\).
2. Given that \(a \neq 0\), find the inverse matrix \(\mathbf { M } ^ { - 1 }\).
3. Hence or otherwise solve the simultaneous equations $$\begin{array} { r } x + 2 y - z = 1 \\ 2 x + 3 y - z = 2 \\ 2 x - y + z = 0 \end{array}$$
4. Find the value of \(k\) for which the simultaneous equations $$\begin{array} { r } 2 y - z = k \\ 2 x + 3 y - z = 2 \\ 2 x - y + z = 0 \end{array}$$ have solutions.
5. Do the equations in part (iv), with the value of \(k\) found, have a solution for which \(x = z\) ? Justify your answer.

4 The matrix equation \(\left( \begin{array} { r r } 6 & - 2 \\ - 3 & 1 \end{array} \right) \binom { x } { y } = \binom { a } { b }\) represents two simultaneous linear equations in \(x\) and \(y\).

1. Write down the two equations.
2. Evaluate the determinant of \(\left( \begin{array} { r r } 6 & - 2 \\ - 3 & 1 \end{array} \right)\). What does this value tell you about the solution of the equations in part (i)?

1

1. Find the inverse of the matrix \(\mathbf { A } = \left( \begin{array} { l l } 4 & 3 \\ 1 & 2 \end{array} \right)\).
2. Use this inverse to solve the simultaneous equations $$\begin{aligned} 4 x + 3 y & = 5 \\ x + 2 y & = - 4 \end{aligned}$$ showing your working clearly.

3

1. Find the inverse of the matrix $$\left( \begin{array} { r r r } 1 & 1 & a \\ 2 & - 1 & 2 \\ 3 & - 2 & 2 \end{array} \right)$$ where \(a \neq 4\).
   Show that when \(a = - 1\) the inverse is $$\frac { 1 } { 5 } \left( \begin{array} { r r r } 2 & 0 & 1 \\ 2 & 5 & - 4 \\ - 1 & 5 & - 3 \end{array} \right)$$
2. Solve, in terms of \(b\), the following system of equations. $$\begin{aligned} x + y - z & = - 2 \\ 2 x - y + 2 z & = b \\ 3 x - 2 y + 2 z & = 1 \end{aligned}$$
3. Find the value of \(b\) for which the equations $$\begin{aligned} x + y + 4 z & = - 2 \\ 2 x - y + 2 z & = b \\ 3 x - 2 y + 2 z & = 1 \end{aligned}$$ have solutions. Give a geometrical interpretation of the solutions in this case. Section B (18 marks)

3

1. Find the value of \(a\) for which the matrix $$\mathbf { M } = \left( \begin{array} { r r r } 1 & 2 & 3 \\ - 1 & a & 4 \\ 3 & - 2 & 2 \end{array} \right)$$ does not have an inverse.
   Assuming that \(a\) does not have this value, find the inverse of \(\mathbf { M }\) in terms of \(a\).
2. Hence solve the following system of equations. $$\begin{aligned} x + 2 y + 3 z & = 1 \\ - x + 4 z & = - 2 \\ 3 x - 2 y + 2 z & = 1 \end{aligned}$$
3. Find the value of \(b\) for which the following system of equations has a solution. $$\begin{aligned} x + 2 y + 3 z & = 1 \\ - x + 6 y + 4 z & = - 2 \\ 3 x - 2 y + 2 z & = b \end{aligned}$$ Find the general solution in this case and describe the solution geometrically.