Determinant calculation and singularity

3. $$\mathbf { A } = \left( \begin{array} { l l l } 2 & k & 2 \\ 2 & 2 & k \\ 1 & 2 & 2 \end{array} \right) \quad \text { where } k \text { is a constant }$$

1. Determine the values of \(k\) for which \(\mathbf { A }\) is singular. Given that \(\mathbf { A }\) is non-singular,
2. find \(\mathbf { A } ^ { - 1 }\), giving your answer in terms of \(k\).
   3.

3. \(\mathbf { M } = \left( \begin{array} { r r r } 3 & 1 & p \\ 1 & 1 & 2 \\ - 1 & p & 2 \end{array} \right)\) where \(p\) is a real constant (a) Find the exact values of \(p\) for which \(\mathbf { M }\) has no inverse. Given that \(\mathbf { M }\) does have an inverse, (b) find \(\mathbf { M } ^ { - 1 }\) in terms of \(p\).
3. \(\mathbf { M } = \left( \begin{array} { r c c } 3 & 1 & p \\ 1 & 1 & 2 \\ - 1 & p & 2 \end{array} \right)\) where \(p\) is a real constant \includegraphics[max width=\textwidth, alt={}, center]{d7a92540-e36c-4e00-bbe4-b253e09962f8-11_2647_1840_118_111}

1. $$\mathbf { A } = \left( \begin{array} { r r r } - 2 & 1 & - 3 \\ k & 1 & 3 \\ 2 & - 1 & k \end{array} \right) \text {, where } k \text { is a constant }$$ Given that the matrix \(\mathbf { A }\) is singular, find the possible values of \(k\).

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

1. Find the value of the determinant of \(\mathbf { M }\).
2. State, giving a brief reason, whether \(\mathbf { M }\) is singular or non-singular.

1 Find the determinant of the matrix \(\left( \begin{array} { r r r } a & 4 & - 1 \\ 3 & a & 2 \\ a & 1 & 1 \end{array} \right)\).

2. The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { c c c } \lambda & 1 & 14 \\ - 1 & 2 & 8 \\ - 3 & 2 & \lambda \end{array} \right)\), where \(\lambda\) is a real constant.

1. Find an expression for the determinant of \(\mathbf { A }\) in terms of \(\lambda\). Give your answer in the form \(a \lambda ^ { 2 } + b \lambda + c\), where \(a , b , c\) are integers whose values are to be determined.
2. Show that \(\mathbf { A }\) is non-singular for all values of \(\lambda\).

7 In this question you must show detailed reasoning. Matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { c c c } a & - 6 & a - 3 \\ a + 9 & a & 4 \\ 0 & - 13 & a - 1 \end{array} \right)\) where \(a\) is a constant.
Find all possible values of \(a\) for which \(\operatorname { det } \mathbf { A }\) has the same value as it has when \(a = 2\).

3 The surface \(S\) has equation \(z = f ( x , y )\), where \(f ( x , y ) = 4 x ^ { 2 } y - 6 x y ^ { 2 } - \frac { 1 } { 12 } x ^ { 4 }\) for all real values of \(x\) and \(y\). You are given that \(S\) has a stationary point at the origin, \(O\), and a second stationary point at the point \(P ( a , b , c )\), where \(\mathrm { c } = \mathrm { f } ( \mathrm { a } , \mathrm { b } )\).

1. Determine the values of \(a , b\) and \(c\).
2. Throughout this part, take the values of \(a\) and \(b\) to be those found in part (a).
   1. Evaluate \(\mathrm { f } _ { x }\) at the points \(\mathrm { U } _ { 1 } ( \mathrm { a } - 0.1 , \mathrm {~b} , \mathrm { f } ( \mathrm { a } - 0.1 , \mathrm {~b} ) )\) and \(\mathrm { U } _ { 2 } ( \mathrm { a } + 0.1 , \mathrm {~b} , \mathrm { f } ( \mathrm { a } + 0.1 , \mathrm {~b} ) )\).
   2. Evaluate \(\mathrm { f } _ { y }\) at the points \(\mathrm { V } _ { 1 } ( \mathrm { a } , \mathrm { b } - 0.1 , \mathrm { f } ( \mathrm { a } , \mathrm { b } - 0.1 ) )\) and \(\mathrm { V } _ { 2 } ( \mathrm { a } , \mathrm { b } + 0.1 , \mathrm { f } ( \mathrm { a } , \mathrm { b } + 0.1 ) )\).
   3. Use the answers to parts (b)(i) and (b)(ii) to sketch the portions of the sections of \(S\), given by
      • \(z = f ( x , b )\), for \(| x - a | \leqslant 0.1\),
3. \(z = f ( a , y )\), for \(| y - b | \leqslant 0.1\).

2 The surface \(S\) is given by \(z = x ^ { 2 } + 4 x y\) for \(- 6 \leqslant x \leqslant 6\) and \(- 2 \leqslant y \leqslant 2\).

1. 1. Write down the equation of any one section of \(S\) which is parallel to the \(x\)-z plane
   2. Sketch the section of (a)(i) on the axes provided in the Printed Answer Booklet.
2. Write down the equation of any one contour of \(S\) which does not include the origin.

5 A trading company deals in two goods. The formula used to estimate \(z\), the total weekly cost to the company of trading the two goods, in tens of thousands of pounds, is \(z = 0.9 x + \frac { 0.096 y } { x } - x ^ { 2 } y ^ { 2 }\),
where \(x\) and \(y\) are the masses, in thousands of tonnes, of the two goods. You are given that \(x > 0\) and \(y > 0\).

1. In the first week of trading, it was found that the values of \(x\) and \(y\) corresponded to the stationary value of \(z\). Determine the total cost to the company for this week.
2. For the second week, the company intends to make a small change in either \(x\) or \(y\) in order to reduce the total weekly cost. Determine whether the company should change \(x\) or \(y\). (You are not expected to say by how much the company should reduce its costs.)