← Further Maths syllabus

SingaporeFurther Maths

Matrices and Linear Spaces

5 dot points across 5 inquiry questions. Click any dot point for a focused answer with worked past exam questions where available.

How do we diagonalise a matrix, and why does diagonalisation make computing powers of a matrix easy?

Diagonalise a matrix using its eigenvalues and eigenvectors and use the diagonal form to compute powers of the matrix
A focused answer to the H2 Further Mathematics outcome on diagonalisation. Writing A = PDP inverse, the conditions for diagonalisability, computing high powers via A^n = PD^nP inverse, and applications to recurrences and systems.
11 min answer →

What are the eigenvalues and eigenvectors of a matrix, and how do we find them?

Find the eigenvalues and eigenvectors of 2x2 and 3x3 matrices using the characteristic equation
A focused answer to the H2 Further Mathematics outcome on eigenvalues and eigenvectors. The eigenvalue equation, the characteristic polynomial, finding eigenvalues from det(A - lambda I) = 0, solving for eigenvectors, and their geometric meaning as invariant directions.
11 min answer →

How do we invert a matrix and use it to solve a system of linear equations?

Find the inverse of a non-singular matrix and use matrices to solve systems of linear equations, recognising consistent, inconsistent and dependent cases
A focused answer to the H2 Further Mathematics outcome on inverse matrices and linear systems. The 2x2 inverse formula, the adjugate method and row reduction for 3x3, solving systems by the inverse, and classifying consistent, inconsistent and dependent systems.
11 min answer →

What is a vector space, and how do the ideas of linear independence, basis, dimension and rank organise it?

Use the concepts of vector spaces and subspaces, linear independence, spanning sets, basis, dimension and the rank of a matrix
A focused answer to the H2 Further Mathematics outcome on linear spaces. Vector spaces and subspaces, linear independence, spanning sets, basis and dimension, the column and null spaces of a matrix, and the rank-nullity relationship.
11 min answer →

How do we perform matrix arithmetic and compute determinants, and what does a determinant tell us?

Carry out matrix addition and multiplication and evaluate the determinant of 2x2 and 3x3 matrices, interpreting its geometric meaning
A focused answer to the H2 Further Mathematics outcome on matrix arithmetic and determinants. Matrix addition and multiplication, non-commutativity, the determinant of 2x2 and 3x3 matrices by cofactor expansion, its properties, and its geometric meaning as an area or volume scale factor.
11 min answer →