a) Consider the following matrices: A=\left[\begin{array}{ccc} -1 & 0 & 1 \\ 0 & 1 & 2 \\ -1 & 1 & 2 \end{array}\right] \quad B=\left[\begin{array}{cc} 1 & 1 \\

2 & -1 \\ -1 & 0 \end{array}\right] \quad C=\left[\begin{array}{cc} 3 & -2 \\ -1 & -2 \end{array}\right] \quad D=\left[\begin{array}{cc} 2 & -4 \\ -1 & -1 \end{array}\right] \quad E=\left[\begin{array}{ll} 3 & 0 \\ 0 & 3 \end{array}\right] i) Compute the product AB if possible and show your working. If not possible, clearly explain why. ii) Compute ((CT+D)"D)" if possible and show all your working. If not possible, clearly explain why. \text { iii) Solve for matrix } F(\text { i.e. find matrix } F), \text { where: }\left\langle\left(E^{\top}-2 \mid z\right)^{-1}\right)^{2}+F^{-1}=\left[\begin{array}{cc} 2 & -1 \\ 0 & 2 \end{array}\right] \text { iv) Let } G=\left[\begin{array}{cc} -2 & 4 \\ 3 & -6 \end{array}\right] \text { . Construct a } 2 \times 2 \text { matrix } \mathbf{G} \text { ' such that the product } \mathrm{GG}^{\prime} \text { is the zero } matrix. Use two different, nonzero columns for G'. Explain how you arrive at your answer. Your answer should not be given by using systems of equations. b) Answer the following: i) Calculate a unit vector that points to the opposite direction to (w-u), where u = (3, -2, 1), w = (3, -2, -2). ii) Consider the vectors u =(1, 2), v= (0, 3) and W =(a, b). Find a, b such that: w is orthogonal to u AND - (W+u) is orthogonal to v. c) In the following attempt to simplify the matrix expression given in Step 0, there are some errors. For each Step 1- 6, identify all errors (if any) and for each error you identify, give an explanation of why it is an error and how it should have been in its correct form.You should consider each step in relation only to the step before as given, e.g. Step 3 based on Step 2 as given, and not based on the corrected version of the previous step. \text { Assume that all matrices involved are invertible. Assume that } A=\left[\begin{array}{ll} 1 & 3 \\ 3 & 1 \end{array}\right] \text { . }