2 The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by \(\mathbf { A } = \left( \begin{array} { l l } 1 & a
3 & 0 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { l l } 4 & 2
3 & 3 \end{array} \right)\).
- Find the value of \(a\) such that \(\mathbf { A B } = \mathbf { B A }\).
- Prove by counter example that matrix multiplication for \(2 \times 2\) matrices is not commutative.
- A triangle of area 4 square units is transformed by the matrix B. Find the area of the image of the triangle following this transformation.
- Find the equations of the invariant lines of the form \(y = m x\) for the transformation represented by matrix \(\mathbf { B }\).