Singular matrix conditions

3. The matrix \(\mathbf { M }\) is defined by $$\mathbf { M } = \left( \begin{array} { c c } k + 5 & - 2 \\ - 3 & k \end{array} \right)$$

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

1. 1. The matrix \(\mathbf { A }\) is defined by

$$\mathbf { A } = \left( \begin{array} { c c } 3 k & 4 k - 1 \\ 2 & 6 \end{array} \right)$$ where \(k\) is a constant.

1. Determine the value of \(k\) for which \(\mathbf { A }\) is singular. Given that \(\mathbf { A }\) is non-singular,
2. determine \(\mathbf { A } ^ { - 1 }\) in terms of \(k\), giving your answer in simplest form.
   (ii) The matrix \(\mathbf { B }\) is defined by $$\mathbf { B } = \left( \begin{array} { l l } p & 0 \\ 0 & q \end{array} \right)$$ where \(p\) and \(q\) are integers.
   State the value of \(p\) and the value of \(q\) when \(\mathbf { B }\) represents
3. an enlargement about the origin with scale factor - 2
4. a reflection in the \(y\)-axis.

6. \(\mathbf { X } = \left( \begin{array} { l l } 1 & a \\ 3 & 2 \end{array} \right)\), where \(a\) is a constant.

1. Find the value of \(a\) for which the matrix \(\mathbf { X }\) is singular. $$\mathbf { Y } = \left( \begin{array} { r r } 1 & - 1 \\ 3 & 2 \end{array} \right)$$
2. Find \(\mathbf { Y } ^ { - 1 }\). The transformation represented by \(\mathbf { Y }\) maps the point \(A\) onto the point \(B\).
   Given that \(B\) has coordinates ( \(1 - \lambda , 7 \lambda - 2\) ), where \(\lambda\) is a constant,
3. find, in terms of \(\lambda\), the coordinates of point \(A\).

7. \(\mathbf { A } = \left( \begin{array} { r r } a & - 2 \\ - 1 & 4 \end{array} \right)\), where \(a\) is a constant.

1. Find the value of \(a\) for which the matrix \(\mathbf { A }\) is singular. $$\mathbf { B } = \left( \begin{array} { r r } 3 & - 2 \\ - 1 & 4 \end{array} \right)$$
2. Find \(\mathbf { B } ^ { - 1 }\). The transformation represented by \(\mathbf { B }\) maps the point \(P\) onto the point \(Q\).
   Given that \(Q\) has coordinates \(( k - 6,3 k + 12 )\), where \(k\) is a constant,
3. show that \(P\) lies on the line with equation \(y = x + 3\).

2. \(\mathbf { M } = \left( \begin{array} { c c } 2 a & 3 \\ 6 & a \end{array} \right)\), where \(a\) is a real constant.

1. Given that \(a = 2\), find \(\mathbf { M } ^ { - 1 }\).
2. Find the values of \(a\) for which \(\mathbf { M }\) is singular.

2. (a) Given that $$\mathbf { A } = \left( \begin{array} { l l l } 3 & 1 & 3 \\ 4 & 5 & 5 \end{array} \right) \quad \text { and } \quad \mathbf { B } = \left( \begin{array} { r r } 1 & 1 \\ 1 & 2 \\ 0 & - 1 \end{array} \right)$$ find \(\mathbf { A B }\).
(b) Given that $$\mathbf { C } = \left( \begin{array} { l l } 3 & 2 \\ 8 & 6 \end{array} \right) , \quad \mathbf { D } = \left( \begin{array} { r r } 5 & 2 k \\ 4 & k \end{array} \right) , \text { where } k \text { is a constant }$$ and $$\mathbf { E } = \mathbf { C } + \mathbf { D }$$ find the value of \(k\) for which \(\mathbf { E }\) has no inverse.

1. $$\mathbf { M } = \left( \begin{array} { c c } x & x - 2 \\ 3 x - 6 & 4 x - 11 \end{array} \right)$$ Given that the matrix \(\mathbf { M }\) is singular, find the possible values of \(x\).

$$\mathbf { A } = \left( \begin{array} { r r } 5 k & 3 k - 1 \\ - 3 & k + 1 \end{array} \right) , \text { where } k \text { is a real constant. }$$ Given that \(\mathbf { A }\) is a singular matrix, find the possible values of \(k\).
(ii) $$\mathbf { B } = \left( \begin{array} { l l } 10 & 5 \\ - 3 & 3 \end{array} \right)$$ A triangle \(T\) is transformed onto a triangle \(T ^ { \prime }\) by the transformation represented by the matrix \(\mathbf { B }\). The vertices of triangle \(T ^ { \prime }\) have coordinates \(( 0,0 ) , ( - 20,6 )\) and \(( 10 c , 6 c )\), where \(c\) is a positive constant. The area of triangle \(T ^ { \prime }\) is 135 square units.

1. Find the matrix \(\mathbf { B } ^ { - 1 }\)
2. Find the coordinates of the vertices of the triangle \(T\), in terms of \(c\) where necessary.
3. Find the value of \(c\).

1. Given that \(k\) is a real number and that

$$\mathbf { A } = \left( \begin{array} { c c } 1 + k & k \\ k & 1 - k \end{array} \right)$$ find the exact values of \(k\) for which \(\mathbf { A }\) is a singular matrix. Give your answers in their simplest form.
(3)

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

1. Find A-4I.
2. Given that \(\mathbf { A }\) is singular, find the value of \(a\).

12

1. Show that the matrix \(\left[ \begin{array} { c c } 5 - k & 2 \\ k ^ { 3 } + 1 & k \end{array} \right]\) is singular when \(k = 1\).
   12
2. Find the values of \(k\) for which the matrix \(\left[ \begin{array} { c c } 5 - k & 2 \\ k ^ { 3 } + 1 & k \end{array} \right]\) has a negative determinant. Fully justify your answer. \(13 \frac { \text { The graph of the rational function } y = \mathrm { f } ( x ) \text { intersects the } x \text {-axis exactly once at } } { ( - 3,0 ) }\) The graph has exactly two asymptotes, \(y = 2\) and \(x = - 1\)

1 Which of the following matrices is singular?
Circle your answer. \(\left[ \begin{array} { l l } 1 & 0 \\ 0 & 1 \end{array} \right]\) \(\left[ \begin{array} { l l } 1 & 1 \\ 2 & 2 \end{array} \right]\) \(\left[ \begin{array} { l l } 0 & 1 \\ 1 & 0 \end{array} \right]\) \(\left[ \begin{array} { c c } 1 & - 2 \\ 1 & 2 \end{array} \right]\)