Linear combinations of $2 \times 2$ matrices. Three examples.

$A,\;B$ $2 \times 2$ matrices. Find eigenvalues and eigenvectors of both. Hence or otherwise, find $B^n$ for largish $n$.

Example of an explore mode question. Student is given a 3x3 matrix and is asked to find the characteristic polynomial and eigenvalues, and then eigenvectors for each eigenvalue. The part asking for eigenvectors can be repeated as often as the student wants, to be used for different eigenvalues.

Assessed: calculating characteristic polynomial and eigenvectors.

Feature: any correct eigenvalue will be recognised by the marking algorithm, even multiples of the obvious one(s) (which can be read off from the reduced row echelon form)

Randomisation: Not randomised, just using particular matrices. I am still working on how to randomise this for 3x3; a randomised 2x2 version exists. I have several different versions for 3x3 (not all published yet), so I could make a random choice between these in a test.

The implementation uses linear algebra functions such as "find reduced echelon form" or "find kernel of a reduced echelon form", from the extension "linalg2".

This exam test students understanding of determinants and inverses of 3x3 matrices.

This exam looks at the determinatns and inverses of 2x2 matrices

Exercises on calculating the determinant of 2x2 and 3x3 matrices.

Exercises on calculating the determinant of 3x3 matrices.

Find the determinant and inverse of three $2 \times 2$ invertible matrices.

This exam gives an overview of skills studnets require to be able to add and subtract matrices.

Exam to test students understanding of matrices multiplied by a scalar.

Multiplication of two matrices.

This question tests if a students understands when matrices are conformable

This question tests understanding of subscript notation for matrices

Adding and subtracting two 3x3 matrices.

Multiplication of $2 \times 2$ matrices.

Addition, subtraction and multiplication by a scalar for 2 x 2 matrices.

Multiplication and adding matrices.

2x2 by ad-bc, 3x3 by Laplace expansion, properties of determinants

Multiplication of square and non square matrices, Identity matrices and zero matrices

Multiplication of $2 \times 2$ matrices.

Determinant of n x m matrix by Laplace Expansion across top row.

Addition, subtraction and multiplication by a scalar for 2 x 2 matrices.

Multiplication of $2 \times 2$ matrices.