Solving 3×3 systems using inverse

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 \(A\).

7. $$\mathbf { A } ( x ) = \left( \begin{array} { c c c } 1 & x & - 1 \\ 3 & 0 & 2 \\ 1 & 1 & 0 \end{array} \right) , x \neq \frac { 5 } { 2 }$$

1. Calculate the inverse of \(\mathbf { A } ( x )\). $$\mathbf { B } = \left( \begin{array} { c c c } 1 & 3 & - 1 \\ 3 & 0 & 2 \\ 1 & 1 & 0 \end{array} \right)$$ The image of the vector \(\left( \begin{array} { c } p \\ q \\ r \end{array} \right)\) when transformed by \(\mathbf { B }\) is \(\left( \begin{array} { l } 2 \\ 3 \\ 4 \end{array} \right)\)
2. Find the values of \(p , q\) and \(r\).

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 $$\begin{aligned} a x + 2 y & = 3 \\ 3 x + y + 2 z & = 4 \\ - y + z & = 1 \end{aligned}$$

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.

10 You are given that \(\mathbf { A } = \left( \begin{array} { r r r } 1 & - 2 & k \\ 2 & 1 & 2 \\ 3 & 2 & - 1 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { r c c } - 5 & - 2 + 2 k & - 4 - k \\ 8 & - 1 - 3 k & - 2 + 2 k \\ 1 & - 8 & 5 \end{array} \right)\) and that \(\mathbf { A B }\) is of the form \(\mathbf { A B } = \left( \begin{array} { c c c } k - n & 0 & 0 \\ 0 & k - n & 0 \\ 0 & 0 & k - n \end{array} \right)\).

1. Find the value of \(n\).
2. Write down the inverse matrix \(\mathbf { A } ^ { - 1 }\) and state the condition on \(k\) for this inverse to exist.
3. Using the result from part (ii), or otherwise, solve the following simultaneous equations. $$\begin{aligned} x - 2 y + z = & 1 \\ 2 x + y + 2 z = & 12 \\ 3 x + 2 y - z = & 3 \end{aligned}$$

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

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

4 You are given that if \(\mathbf { M } = \left( \begin{array} { r r r } 4 & 0 & 1 \\ - 6 & 1 & 1 \\ 5 & 2 & 5 \end{array} \right)\) then \(\mathbf { M } ^ { - 1 } = \frac { 1 } { k } \left( \begin{array} { r r r } - 3 & - 2 & 1 \\ - 35 & - 15 & 10 \\ 17 & 8 & - 4 \end{array} \right)\).
Find the value of \(k\). Hence solve the following simultaneous equations. $$\begin{aligned} 4 x + z & = 9 \\ - 6 x + y + z & = 32 \\ 5 x + 2 y + 5 z & = 81 \end{aligned}$$

3 You are given that \(\mathbf { N } = \left( \begin{array} { r r r } - 9 & - 2 & - 4 \\ 3 & 2 & 2 \\ 5 & 1 & 2 \end{array} \right)\) and \(\mathbf { N } ^ { - 1 } = \left( \begin{array} { r r r } 1 & 0 & 2 \\ 2 & 1 & 3 \\ - \frac { 7 } { 2 } & p & - 6 \end{array} \right)\).

1. Find the value of \(p\).
2. Solve the equation \(\mathbf { N } \left( \begin{array} { c } x \\ y \\ z \end{array} \right) = \left( \begin{array} { r } - 39 \\ 5 \\ 22 \end{array} \right)\).

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

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.

4

1. Find \(\mathbf { M } ^ { - 1 }\), where \(\mathbf { M } = \left( \begin{array} { r r r } 1 & 2 & 3 \\ - 1 & 1 & 2 \\ - 2 & 1 & 2 \end{array} \right)\).
2. Hence find, in terms of the constant \(k\), the point of intersection of the planes $$\begin{aligned} x + 2 y + 3 z & = 19 \\ - x + y + 2 z & = 4 \\ - 2 x + y + 2 z & = k \end{aligned}$$
3. In this question you must show detailed reasoning. Find the acute angle between the planes \(x + 2 y + 3 z = 19\) and \(- x + y + 2 z = 4\).

4. Solve the simultaneous equations $$\begin{aligned} 4 x - 2 y + 3 z & = 8 \\ 2 x - 3 y + 8 z & = - 1 \\ 2 x + 4 y - z & = 0 \end{aligned}$$

6. The matrix \(\mathbf { M }\) is given by $$\mathbf { M } = \left[ \begin{array} { l l l } 2 & 1 & 3 \\ 1 & 3 & 2 \\ 3 & 2 & 5 \end{array} \right]$$

1. Find
   1. the adjugate matrix of \(\mathbf { M }\),
   2. hence determine the inverse matrix \(\mathbf { M } ^ { - 1 }\).
2. Use your result to solve the simultaneous equations $$\begin{aligned} & 2 x + y + 3 z = 13 \\ & x + 3 y + 2 z = 13 \\ & 3 x + 2 y + 5 z = 22 \end{aligned}$$

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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8. $$\mathbf { A } = \left( \begin{array} { r r r } 3 & 1 & - 1 \\ 1 & 1 & 1 \\ k & 3 & 6 \end{array} \right) \quad k \neq 0$$

1. Find, in terms of \(k , \mathbf { A } ^ { - 1 }\)
2. Determine, in simplest form in terms of \(k\), the coordinates of the point where the following planes intersect. $$\begin{array} { r } 3 x + y - z = 3 \\ x + y + z = 1 \\ k x + 3 y + 6 z = 6 \end{array}$$

1 The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { c c c } - 3 & 3 & 2 \\ 5 & - 4 & - 3 \\ - 1 & 1 & 1 \end{array} \right)\).

1. Find \(\mathbf { A } ^ { - 1 }\).
2. Solve the simultaneous equations $$\begin{aligned} - 3 x + 3 y + 2 z & = 12 a \\ 5 x - 4 y - 3 z & = - 6 \\ - x + y + z & = 7 \end{aligned}$$ giving your solution in terms of \(a\).

5 \end{array} \right) + \lambda \left( \begin{array} { l } 1
1
3 \end{array} \right) + \mu \left( \begin{array} { c } - 1
2
1 \end{array} \right) .$$

1. Find a vector which is perpendicular to \(\Pi\).
2. Hence find an equation for \(\Pi\) in the form r.n \(= p\).
3. Find in the form \(\sqrt { q }\) the shortest distance between \(\Pi\) and the origin, where \(q\) is a rational number. 4 The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { c r c } a & 2 & 3 \\ 4 & 4 & 6 \\ - 2 & 2 & 9 \end{array} \right)\) where \(a\) is a constant. It is given that if \(\mathbf { A }\) is not singular then $$\mathbf { A } ^ { - 1 } = \frac { 1 } { 24 a - 48 } \left( \begin{array} { c c c } 24 & - 12 & 0
   - 48 & 9 a + 6 & 12 - 6 a
   16 & - 2 a - 4 & 4 a - 8 \end{array} \right)$$
4. Use \(\mathbf { A } ^ { - 1 }\) to solve the simultaneous equations below, giving your answer in terms of \(k\). $$\begin{array} { r } x + 2 y + 3 z = 6
   4 x + 4 y + 6 z = 8
   - 2 x + 2 y + 9 z = k \end{array}$$
5. Consider the equations below where \(a\) takes the value which makes \(\mathbf { A }\) singular. $$\begin{aligned} a x + 2 y + 3 z & = b
   4 x + 4 y + 6 z & = 10
   - 2 x + 2 y + 9 z & = - 13 \end{aligned}$$ \(b\) takes the value for which the equations have an infinite number of solutions.
   • Determine the value of \(b\).
   • Find the solutions for \(y\) and \(z\) in terms of \(x\).
   • For the equations in part (ii) with the values of \(a\) and \(b\) found in part (ii) describe fully the geometrical arrangement of the planes represented by the equations.
   5 The region \(R\) between the \(x\)-axis, the curve \(y = \frac { 1 } { \sqrt { p + x ^ { 2 } } }\) and the lines \(x = \sqrt { p }\) and \(x = \sqrt { 3 p }\), where \(p\) is a positive parameter, is rotated by \(2 \pi\) radians about the \(x\)-axis to form a solid of revolution \(S\).
6. Find and simplify an algebraic expression, in terms of \(p\), for the exact volume of \(S\).
7. Given that \(R\) must lie entirely between the lines \(x = 1\) and \(x = \sqrt { 48 }\) find in exact form
   • the greatest possible value of the volume of \(S\)
   • the least possible value of the volume of \(S\).

5 You are given that \(\mathbf { A } = \left( \begin{array} { c c c } 1 & 2 & 1 \\ 2 & 5 & 2 \\ 3 & - 2 & - 1 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { c c c } 1 & 0 & 1 \\ - 8 & 4 & 0 \\ 19 & - 8 & - 1 \end{array} \right)\).

1. Find \(\mathbf { A B }\).
2. Hence write down \(\mathbf { A } ^ { - 1 }\).
3. You are given three simultaneous equations $$\begin{array} { r } x + 2 y + z = 0 \\ 2 x + 5 y + 2 z = 1 \\ 3 x - 2 y - z = 4 \end{array}$$
   1. Explain how you can tell, without solving them, that there is a unique solution to these equations.
   2. Find this unique solution.

19. $$\mathbf { A } = \left( \begin{array} { r r r } 3 & 1 & - 1 \\ 1 & 1 & 1 \\ 5 & 3 & u \end{array} \right) , \quad u \neq 1$$

1. Show that \(\operatorname { det } \mathbf { A } = 2 ( u - 1 )\).
2. Find the inverse of \(\mathbf { A }\). The image of the vector \(\left( \begin{array} { l } a \\ b \\ c \end{array} \right)\) when transformed by the matrix \(\left( \begin{array} { r r r } 3 & 1 & - 1 \\ 1 & 1 & 1 \\ 5 & 3 & 6 \end{array} \right)\) is \(\left( \begin{array} { l } 3 \\ 1 \\ 6 \end{array} \right)\).
3. Find the values of \(a , b\) and \(c\).
   (3)
   [0pt] [P6 June 2003 Qn 6]

12 The matrix \(\mathbf { A } = \left[ \begin{array} { c c c } 1 & 5 & 3 \\ 4 & - 2 & p \\ 8 & 5 & - 11 \end{array} \right]\), where \(p\) is a constant.
12

1. Given that \(\mathbf { A }\) is a non-singular matrix, find \(\mathbf { A } ^ { - 1 }\) in terms of \(p\).
   State any restrictions on the value of \(p\).
   12
2. The equations below represent three planes. $$\begin{aligned} x + 5 y + 3 z & = 5 \\ 4 x - 2 y + p z & = 24 \\ 8 x + 5 y - 11 z & = - 30 \end{aligned}$$ 12
3. 1. Find, in terms of \(p\), the coordinates of the point of intersection of the three planes.
      [0pt] [4 marks]
      12
4. (ii) In the case where \(p = 2\), show that the planes are mutually perpendicular.

7

1. 1. Given that \(\mathbf { M }\) is a non-singular matrix, find \(\mathbf { M } ^ { - 1 }\) in terms of \(k\) 7
2. (ii) State any restrictions on the value of \(k\) 7
3. Using your answer to part (a)(i), solve $$\begin{array} { r } x + 7 y - 3 z = 6 \\ 3 x + 6 y + 6 z = 3 \\ x + 3 y + 2 z = 1 \end{array}$$