7 You are given that \(a\) is a parameter which can take only real values.
The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { c c r } 2 & 4 & - 6
- 3 & 10 - 4 a & 9
7 & 4 & 4 \end{array} \right)\).
- 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 \(\mathbf { A } \left( \begin{array} { c } \mathrm { x }
\mathrm { y }
\mathrm { z } \end{array} \right) = \left( \begin{array} { r } 6
- 9
11 \end{array} \right)\). - In the case where \(\mathbf { A }\) is not singular, solve the given system of equations by using \(\mathbf { A } ^ { - 1 }\).
- 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.
- Determine the range of values of \(a\) for which the orientation of the image is the reverse of the orientation of the object.
- Determine the range of values of \(a\) for which the volume of the image is less than the volume of the object.