Describe reflection from matrix

2. $$\mathbf { A } = \left( \begin{array} { l l } 2 & 0 \\ 5 & 3 \end{array} \right) , \quad \mathbf { B } = \left( \begin{array} { r r } - 3 & - 1 \\ 5 & 2 \end{array} \right)$$

1. Find \(\mathbf { A B }\). Given that $$\mathbf { C } = \left( \begin{array} { r r } - 1 & 0 \\ 0 & 1 \end{array} \right)$$
2. describe fully the geometrical transformation represented by \(\mathbf { C }\),
3. write down \(\mathbf { C } ^ { 100 }\). \includegraphics[max width=\textwidth, alt={}, center]{d20fa710-2d91-4ac2-adbc-46ccdcb93380-03_99_97_2631_1784}

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1. The transformation P is represented by the matrix \(\left( \begin{array} { r r } 1 & 0 \\ 0 & - 1 \end{array} \right)\). Give a geometrical description of transformation P .
2. The transformation Q is represented by the matrix \(\left( \begin{array} { r r } 0 & - 1 \\ - 1 & 0 \end{array} \right)\). Give a geometrical description of transformation Q.
3. The transformation R is equivalent to transformation P followed by transformation Q . Find the matrix that represents R .
4. Give a geometrical description of the single transformation that is represented by your answer to part (iii).

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1. The transformation T is defined by the matrix \(\mathbf { A }\), where $$\mathbf { A } = \left[ \begin{array} { r r } 0 & - 1 \\ - 1 & 0 \end{array} \right]$$
   1. Describe the transformation T geometrically.
   2. Calculate the matrix product \(\mathbf { A } ^ { 2 }\).
   3. Explain briefly why the transformation T followed by T is the identity transformation.
2. The matrix \(\mathbf { B }\) is defined by $$\mathbf { B } = \left[ \begin{array} { l l } 1 & 1 \\ 0 & 1 \end{array} \right]$$
   1. Calculate \(\mathbf { B } ^ { 2 } - \mathbf { A } ^ { 2 }\).
   2. Calculate \(( \mathbf { B } + \mathbf { A } ) ( \mathbf { B } - \mathbf { A } )\).

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1. The matrix \(\mathbf { M }\) is \(\left( \begin{array} { r r } 0 & - 1 \\ - 1 & 0 \end{array} \right)\).
   1. Find \(\mathbf { M } ^ { 2 }\).
   2. Write down the transformation represented by \(\mathbf { M }\).
   3. Hence state the geometrical significance of the result of part (i).
2. The matrix \(\mathbf { N }\) is \(\left( \begin{array} { c c } k + 1 & 0 \\ k & k + 2 \end{array} \right)\), where \(k\) is a constant. Using determinants, investigate whether \(\mathbf { N }\) can represent a reflection.

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1. Specify fully the transformations represented by the following matrices.