1 In this question you must show detailed reasoning. Use an algebraic method to find the square roots of \(- 77 - 36 \mathrm { i }\).
\(2 \mathrm { P } , \mathrm { Q }\) and T are three transformations in 2-D.
P is a reflection in the \(x\)-axis. \(\mathbf { A }\) is the matrix that represents P .

1. Write down the matrix \(\mathbf { A }\). Q is a shear in which the \(y\)-axis is invariant and the point \(\binom { 1 } { 0 }\) is transformed to the point \(\binom { 1 } { 2 }\). \(\mathbf { B }\) is the
   matrix that represents Q . matrix that represents Q.
2. Find the matrix \(\mathbf { B }\). T is P followed by Q. C is the matrix that represents T.
3. Determine the matrix \(\mathbf { C }\).
   \(L\) is the line whose equation is \(y = x\).
4. Explain whether or not \(L\) is a line of invariant points under \(T\). An object parallelogram, \(M\), is transformed under T to an image parallelogram, \(N\).
5. Explain what the value of the determinant of \(\mathbf { C }\) means about
   • the area of \(N\) compared to the area of \(M\),
   • the orientation of \(N\) compared to the orientation of \(M\).