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

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

Cofactors Determinant and inverse of a 3x3 matrix.

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 question tests learner's knowledge of the inverse matrix method for a 3x3 matrix.

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

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

Example of an explore mode question. Student is given a 2x2 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 eigenvector is recognised by the marking algorithm, also multiples of the "obvious" one(s) (given the reduced row echelon form that we use to calculate them).

Randomisation: a random true/false for invertibility is created, and the eigenvalues a and b are randomised (condition: two different evalues, and a=0 iff invertibility is false), and a random invertible 2x2 matrix with determinant 1 or -1 is created (via random elementary row operations) to change base from diag(a,b) to the matrix for the question. Determinant 1 or -1 ensures that we keep integer entries.

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

Cofactors Determinant and inverse of a 3x3 matrix.

Find the determinant of a $3 \times 3$ matrix.

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

Quiz to assess matrix addition, subtraction, multiplication and multiplication by scalar, determinants and inverses, solving a system of simultaneous equations.

Find the determinant of a $4 \times 4$ matrix.

Find the determinant of a $3 \times 3$ matrix.

Example of an explore mode question. Student is given a 2x2 matrix with eigenvalues and eigenvectors, and is asked to decide if the matrix is invertible. If yes, second and third parts are offered where the student should give the eigenvalues and eigenvectors of the inverse matrix.

Assessed: remembering link between 0 eigenvalue and invertibility. Remembering link between eigenvalues and eigenvectors of matrix and its inverse.

Randomisation: a random true/false for invertibility is created, and the eigenvalues a and b are randomised (condition: two different evalues, and a=0 iff invertibility is false), and a random invertible 2x2 matrix with determinant 1 or -1 is created (via random elementary row operations) to change base from diag(a,b) to the matrix for the question. Determinant 1 or -1 ensures that we keep integer entries.

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

Determinant of 2x2 and notation

Input answer only.

Determinant of 2x2 and notation

Students are asked to form the calculation before giving answer.

Useful properties of determinants that allow simplification.

Find the determinant of a $3 \times 3$ matrix.

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

Quiz to assess matrix addition, subtraction, multiplication and multiplication by scalar, determinants and inverses, solving a system of simultaneous equations.

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

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

Matrix addition, multiplication. Finding inverse. Determinants. Systems of equations.

rebelmaths

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

Nature of fixed points of a 2D dynamical system.

These examples are either centres or spirals.

Cofactors Determinant and inverse of a 3x3 matrix.

Semi-worked example of inverting a 3x3 matrix using cofactor, adjoint and determinant.