Matrices – Further Pure Maths 2

Candidates should be able to:

• formulate a problem involving the solution of 3 linear simultaneous equations in 3 unknowns as a problem involving the solution of a matrix equation, or vice versa
• understand the cases that may arise concerning the consistency or inconsistency of 3 linear simultaneous equations, relate them to the singularity or otherwise of the corresponding matrix, solve consistent systems, and interpret geometrically in terms of lines and planes (e.g. three planes meeting in a common point, or in a common line, or having no common points.)
• understand the terms ‘characteristic equation’, ‘eigenvalue’ and ‘eigenvector’, as applied to square matrices (Including use of the definition \(Ae = \lambda e\) to prove simple properties, e.g. that \(\lambda^n\) is an eigenvalue of \(A^n\).)
• find eigenvalues and eigenvectors of 2 × 2 and 3 × 3 matrices (Restricted to cases where the eigenvalues are real and distinct.)
• express a square matrix in the form \(QDQ^{-1}\), where \(D\) is a diagonal matrix of eigenvalues and \(Q\) is a matrix whose columns are eigenvectors, and use this expression (e.g. in calculating powers of 2 × 2 or 3 × 3 matrices.)
• use the fact that a square matrix satisfies its own characteristic equation. (e.g. in finding successive powers of a matrix or finding an inverse matrix; restricted to 2 × 2 or 3 × 3 matrices only.)

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Matrices – Further Pure Maths 2 Video (5 Parts)

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