2013 SQA Advanced Higher Maths no.3 : 2x2 matrices

2x2 matrices.
Square of a matrix with algebraic terms as entries.
Solving the determinant = 0 to find the condition for a singular matrix.
Matrix equation with a transpose.

Instructional exercise consisting of question 3 from
the 2013 SQA Advanced Higher Mathematics exam.
Set out in stages of 'try then check' the worked solution to each part of the question.