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LearnAQA7367First teaching 2017Paper 2 · Compulsory contentMatrices

Paper 2 · Module

Matrices

3 subtopics≈ 140 min in total

Start the module: Matrix algebra and inverses Open graphing calculator

Matrix algebra and inverses
Matrices multiply row-by-column. The determinant measures how a matrix scales area (or volume); when it is zero the matrix has no inverse. For $2\times2$ , $\mathbf A^{-1}=\frac1{ad-bc}\begin{pmatrix}d&-b\\-c&a\end{pmatrix}$ , and inverses solve systems of equations in one step.
≈ 45 min
Transformations, invariant points and lines
The columns of a transformation matrix are the images of the unit vectors $\mathbf i$ and $\mathbf j$ . Rotations, reflections, enlargements and stretches all have standard matrices. $\mathbf{BA}$ means "do $\mathbf A$ , then $\mathbf B$ ". Invariant points satisfy $\mathbf M\mathbf x=\mathbf x$ ; invariant lines map onto themselves.
≈ 45 min
Eigenvalues and diagonalisation
An eigenvector of $\mathbf M$ keeps its direction: $\mathbf M\mathbf v=\lambda\mathbf v$ , where $\lambda$ is the eigenvalue. Eigenvalues solve $\det(\mathbf M-\lambda\mathbf I)=0$ . Putting eigenvectors in $\mathbf P$ and eigenvalues in $\mathbf D$ gives $\mathbf M=\mathbf{PDP}^{-1}$ , so $\mathbf M^n=\mathbf{PD}^n\mathbf P^{-1}$ .
≈ 50 min