4.03k Determinant 3x3: volume scale factor

5. $$\mathbf { M } = \left( \begin{array} { r r r } 6 & - 2 & - 1 \\ - 2 & 6 & - 1 \\ - 1 & - 1 & 5 \end{array} \right)$$ Given that 8 is an eigenvalue of \(\mathbf { M }\)

1. determine an eigenvector corresponding to the eigenvalue 8
2. Determine the other two eigenvalues of \(\mathbf { M }\).
3. Hence find an orthogonal matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { P } ^ { T } \mathbf { M P } = \mathbf { D }\) 5.
   

4. The matrix \(\mathbf { M }\) is given by $$\left( \begin{array} { r r r } 2 & 0 & - 1 \\ k & 3 & 2 \\ - 2 & 1 & k \end{array} \right)$$

1. Show that \(\operatorname { det } \mathbf { M } = 5 k - 10\) Given that \(k \neq 2\)
2. find \(\mathbf { M } ^ { - 1 }\) in terms of \(k\). The points \(O ( 0,0,0 ) , A ( 4 , - 8,3 ) , B ( - 2,5 , - 4 )\) and \(C ( 4 , - 6,8 )\) are the vertices of a tetrahedron \(T\). The transformation represented by matrix \(\mathbf { M }\) transforms \(T\) to a tetrahedron with volume 50
3. Determine the possible values of \(k\).

5

1. Find the volume scale factor of the transformation with associated matrix \(\left( \begin{array} { r r r } 1 & 2 & 0 \\ 0 & 3 & - 1 \\ - 1 & 0 & 2 \end{array} \right)\).
2. The transformations S and T of the plane have associated \(2 \times 2\) matrices \(\mathbf { P }\) and \(\mathbf { Q }\) respectively.
   1. Write down an expression for the associated matrix of the combined transformation S followed by T. The determinant of \(\mathbf { P }\) is 3 and \(\mathbf { Q } = \left( \begin{array} { r r } k & 3 \\ - 1 & 2 \end{array} \right)\), where \(k\) is a constant.
   2. Given that this combined transformation preserves both orientation and area, determine the value of \(k\).

4 The matrix \(\mathbf { M }\) is \(\left( \begin{array} { r r r } 0 & - 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{array} \right)\).

1. 1. Calculate \(\operatorname { det } \mathbf { M }\).
   2. State two geometrical consequences of this value for the transformation associated with \(\mathbf { M }\).
2. Describe fully the transformation associated with \(\mathbf { M }\).

7 A transformation A is represented by the matrix \(\mathbf { A }\) where \(\mathbf { A } = \left( \begin{array} { c c c } - 1 & x & 2 \\ 7 - x & - 6 & 1 \\ 5 & - 5 x & 2 x \end{array} \right)\).
The tetrahedron \(H\) has vertices at \(O , P , Q\) and \(R\). The volume of \(H\) is 6 units. \(P ^ { \prime } , Q ^ { \prime } , R ^ { \prime }\) and \(H ^ { \prime }\) are the images of \(P , Q , R\) and \(H\) under A .

1. In the case where \(x = 5\)

4 A transformation A is represented by the matrix \(\mathbf { A }\) where \(\mathbf { A } = \left( \begin{array} { c c c } - 1 & x & 2 \\ 7 - x & - 6 & 1 \\ 5 & - 5 x & 2 x \end{array} \right)\).
The tetrahedron \(H\) has vertices at \(O , P , Q\) and \(R\). The volume of \(H\) is 6 units. \(P ^ { \prime } , Q ^ { \prime } , R ^ { \prime }\) and \(H ^ { \prime }\) are the images of \(P , Q , R\) and \(H\) under A .

1. In the case where \(x = 5\)

Matrix M is given by \(\mathbf{M} = \begin{pmatrix} k & 1 & -5 \\ 2 & 3 & -3 \\ -1 & 2 & 2 \end{pmatrix}\), where \(k\) is a constant.

1. Show that \(\det \mathbf{M} = 12(k - 3)\). [2]
2. Find a solution of the following simultaneous equations for which \(x \neq z\). $$4x^2 + y^2 - 5z^2 = 6$$ $$2x^2 + 3y^2 - 3z^2 = 6$$ $$-x^2 + 2y^2 + 2z^2 = -6$$ [3]
3. 1. Verify that the point \((2, 0, 1)\) lies on each of the following three planes. $$3x + y - 5z = 1$$ $$2x + 3y - 3z = 1$$ $$-x + 2y + 2z = 0$$ [1]
   2. Describe how the three planes in part (iii) (A) are arranged in 3-D space. Give reasons for your answer. [4]
4. Find the values of \(k\) for which the transformation represented by M has a volume scale factor of 6. [3]

The matrix \(\mathbf{M}\) is defined by $$\mathbf{M} = \begin{pmatrix} 12 & 30 & 8 \\ 18 & 25 & 20 \\ 19 & 50 & 16 \end{pmatrix}.$$

1. Given that \(\det \mathbf{M} = -1040\), give a geometrical interpretation of the solution to the following equation. [2] $$\begin{pmatrix} 12 & 30 & 8 \\ 18 & 25 & 20 \\ 19 & 50 & 16 \end{pmatrix} \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} 2668 \\ 3402 \\ 4581 \end{pmatrix}$$
2. Three hotels A, B, C each have different types of room available to book: single, double and family rooms. For each type of room, the price per night is the same in each of the three hotels. The table below gives, for each hotel, details of the number of each type of room and the total revenue per night when the hotel is full.

   \multirow{2}{*}{Hotel}	Types of room			\multirow{2}{*}{Total revenue}
   \cline{2-4}	Single	Double	Family
   A	12	30	8	£2,668
   B	18	25	20	£3,402
   C	19	50	16	£4,581

   Find the price per night of each type of room. [6]