Matrices and Vectors
Matrices model transformations and simultaneous equations, while vectors describe displacement, direction and geometric relationships.
Learning objectives
After studying these notes, learners should be able to define the main terms, apply the central rules and formulas, solve structured examples and explain each answer clearly.
Key ideas
- Matrix operations and compatibility
- Determinants and inverses
- Solving simultaneous equations with matrices
- Transformation matrices
- Parallel, equal and collinear vectors
- Position vectors and ratios
Essential rules and formulas
- For A = [[a,b],[c,d]], A inverse = 1/(ad-bc)[[d,-b],[-c,a]]
- A vector from P to Q = position(Q) – position(P)
- Parallel vectors are scalar multiples
Worked example
For A = [[2,1],[1,1]], det(A)=1 and A inverse = [[1,-1],[-1,2]].
Study method
Begin every problem by listing the information supplied and identifying the rule that connects it to the unknown quantity. Keep algebraic steps on separate lines, include units where required and check that the final result is reasonable.
Practice questions
- Find the inverse of [[3,2],[1,1]].
- Show that (6,9) and (2,3) are parallel.
- Find the midpoint of points with position vectors (2,5) and (8,-1).
Revision checklist
- Can you state each definition without looking at the notes?
- Can you select the correct formula and substitute values accurately?
- Can you explain the reasoning behind each step?
- Can you solve a similar examination question independently?
