(c) Given matrices
[tex]\[ A=\begin{pmatrix} 1 & 0 & 2 \\ 1 & 1 & 0 \\ 2 & 1 & 1 \end{pmatrix} \][/tex]
and
[tex]\[ B=\begin{pmatrix} 2 & 4 & 1 \\ 0 & 1 & 1 \\ 2 & 3 & 4 \end{pmatrix} \][/tex]

Using the matrices, show that:

i) [tex]\[ (A + B)^T = A^T + B^T \][/tex] (2 marks)

ii) [tex]\[ (AB)^T = B^T A^T \][/tex] (2 marks)

(d) Show that:

i) The Cartesian n-space [tex]\[ \mathbb{R}^n = \{(x_1, x_2, \ldots, x_n) : x_i \in \mathbb{R} \} \][/tex] is a vector space. (2 marks)

ii) Determine if [tex]\[ S = \{(1,2,1), (2,9,0), (3,3,4) \} \][/tex] is a basis for [tex]\(\mathbb{R}^3\)[/tex]. (4 marks)

(e)

i) Show that the Wronskian [tex]\[ W\left(e^x, e^{2x}, 0\right) = 1 \][/tex] (2 marks)

ii) By first partitioning the matrix
[tex]\[ B = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 3 & 2 \\ 1 & 0 & 3 \end{pmatrix} \][/tex]
find the determinant of matrix [tex]\(\beta\)[/tex].