Area transformation under matrices

3. $$\mathbf { A } = \left( \begin{array} { l l } 6 & 4 \\ 1 & 1 \end{array} \right)$$

1. Show that \(\mathbf { A }\) is non-singular. The triangle \(R\) is transformed to the triangle \(S\) by the matrix \(\mathbf { A }\).
   Given that the area of triangle \(R\) is 10 square units,
2. find the area of triangle \(S\). Given that $$\mathbf { B } = \mathbf { A } ^ { 4 }$$ and that the triangle \(R\) is transformed to the triangle \(T\) by the matrix \(\mathbf { B }\),
3. find, without evaluating \(\mathbf { B }\), the area of triangle \(T\).

1. (i) \(\mathbf { A } = \left( \begin{array} { c c } - 3 & 8 \\ - 3 & k \end{array} \right) \quad\) where \(k\) is a constant The transformation represented by \(\mathbf { A }\) transforms triangle \(T\) to triangle \(T ^ { \prime }\) The area of triangle \(T ^ { \prime }\) is three times the area of triangle \(T\)

Determine the possible values of \(k\) (ii) \(\mathbf { B } = \left( \begin{array} { r r } a & - 4 \\ 2 & 3 \end{array} \right)\) and \(\mathbf { B C } = \left( \begin{array} { l l l } 2 & 5 & 1 \\ 1 & 4 & 2 \end{array} \right)\) where \(a\) is a constant Determine, in terms of \(a\), the matrix \(\mathbf { C }\)

8. $$\mathbf { A } = \left( \begin{array} { r r } 2 & - 2 \\ - 1 & 3 \end{array} \right)$$

1. Find \(\operatorname { det } \mathbf { A }\).
2. Find \(\mathbf { A } ^ { - 1 }\). The triangle \(R\) is transformed to the triangle \(S\) by the matrix \(\mathbf { A }\). Given that the area of triangle \(S\) is 72 square units,
3. find the area of triangle \(R\). The triangle \(S\) has vertices at the points \(( 0,4 ) , ( 8,16 )\) and \(( 12,4 )\).
4. Find the coordinates of the vertices of \(R\).

2.

1. $$\mathbf { A } = \left( \begin{array} { c c } - 4 & 10 \\ - 3 & k \end{array} \right) , \quad \text { where } k \text { is a constant. }$$ The triangle \(T\) is transformed to the triangle \(T ^ { \prime }\) by the transformation represented by \(\mathbf { A }\). Given that the area of triangle \(T ^ { \prime }\) is twice the area of triangle \(T\), find the possible values of \(k\).
2. Given that $$\mathbf { B } = \left( \begin{array} { r r r } 1 & - 2 & 3 \\ - 2 & 5 & 1 \end{array} \right) , \quad \mathbf { C } = \left( \begin{array} { r r } 2 & 8 \\ 0 & 2 \\ 1 & - 2 \end{array} \right)$$ find \(\mathbf { B C }\). \includegraphics[max width=\textwidth, alt={}, center]{9093bb1d-4f32-44e7-b0e7-b8c4f8a844e1-05_124_42_2608_1902}

5. \(\mathbf { A } = \left( \begin{array} { r r } - 4 & a \\ b & - 2 \end{array} \right)\), where \(a\) and \(b\) are constants. Given that the matrix \(\mathbf { A }\) maps the point with coordinates \(( 4,6 )\) onto the point with coordinates \(( 2 , - 8 )\),

1. find the value of \(a\) and the value of \(b\). A quadrilateral \(R\) has area 30 square units.
   It is transformed into another quadrilateral \(S\) by the matrix \(\mathbf { A }\).
   Using your values of \(a\) and \(b\),
2. find the area of quadrilateral \(S\).

6. $$\mathbf { M } = \left( \begin{array} { r r } 8 & - 1 \\ - 4 & 2 \end{array} \right)$$

1. Find the value of \(\operatorname { det } \mathbf { M }\) The triangle \(T\) has vertices at the points \(( 4,1 ) , ( 6 , k )\) and \(( 12,1 )\), where \(k\) is a constant.
   The triangle \(T\) is transformed onto the triangle \(T ^ { \prime }\) by the transformation represented by the matrix \(\mathbf { M }\). Given that the area of triangle \(T ^ { \prime }\) is 216 square units,
2. find the possible values of \(k\).

1 You are given the matrix \(\mathbf { M } = \left( \begin{array} { r r } 2 & 3 \\ - 2 & 1 \end{array} \right)\).
Find the inverse of \(\mathbf { M }\).
The transformation associated with \(\mathbf { M }\) is applied to a figure of area 2 square units. What is the area of the transformed figure?

1 You are given the matrix \(\mathbf { M } = \left( \begin{array} { r r } 2 & - 1 \\ 4 & 3 \end{array} \right)\).

1. Find the inverse of \(\mathbf { M }\).
2. A triangle of area 2 square units undergoes the transformation represented by the matrix \(\mathbf { M }\). Find the area of the image of the triangle following this transformation.

2 You are given that \(\mathbf { M } = \left( \begin{array} { r r } 4 & 0 \\ - 1 & 3 \end{array} \right)\).

1. The transformation associated with \(\mathbf { M }\) is applied to a figure of area 3 square units. Find the area of the transformed figure.
2. Find \(\mathbf { M } ^ { - 1 }\) and \(\operatorname { det } \mathbf { M } ^ { - 1 }\).
3. Explain the significance of \(\operatorname { det } \mathbf { M } \times \operatorname { det } \mathbf { M } ^ { - 1 }\) in terms of transformations.

1 The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { c c } 8 & - 2 \\ p & 1 \end{array} \right)\), where \(p \neq - 4\).

1. Find the inverse of \(\mathbf { M }\) in terms of \(p\).
2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{578345cb-e7a1-41fd-abf8-a977912965e8-2_1086_885_584_587} \captionsetup{labelformat=empty} \caption{Fig. 1}
   \end{figure} The triangle shown in Fig. 1 undergoes the transformation represented by the matrix \(\left( \begin{array} { c c } 8 & - 2 \\ 3 & 1 \end{array} \right)\). Find the area of the image of the triangle following this transformation.