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

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

Given a 3 x 3 matrix, and two eigenvectors find their corresponding eigenvalues. Also fnd the characteristic polynomial and using this find the third eigenvalue and a normalised eigenvector $(x=1)$.

Reduce a 5x6 matrix to row reduced form and using this find rank and nullity.

Putting a pair of linear equations into matrix notation and then solving by finding the inverse of the coefficient matrix. 

Student finds a basis for kernel and image of a matrix transformation. Any basis can be entered; there is a custom marking algorithm which checks if it is a correct basis.

There are options to adjust this question fairly easily, for example to get different variants for practice and for a test, by changing the options in the "pivot columns" in the variables. You should be careful to think about and test your pivot options, as some are easier or harder than others, and some don't work very well.

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

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

Matrix multiplication. Has automatically generated "unresolved" matrix product to write in the solution, which is the interesting part of this implementation.

Finding a matrix from a formula for each entry, which involves the row and column numbers of that entry. Randomized size of the matrices and formula.

The interesting part about the implementation is the way the output is generated for "Advice".

Educational calculation tool rather than "question".

This allows the student to input a square matrix (max rows 5). Then the student can decide to swap some rows, or multiply some rows, or add multiples of one row to other rows. The student only has to input what operation should be performed, and this is automatically applied to the matrix and the identity matrix (or what it has got to). This question has no marks and no feedback as it's just meant as a "calculator". It has some checks in so students know when they are not entering a square matrix or a valid row number etc.

It has some rounding to 13 decimal places, as otherwise some fraction calculations become incorrectly displayed as a very small number instead of 0.

It would be possible to extend to more than 5 rows, one just has to put in a lot more variables and so on. I just had to choose some place to stop.

Based on Chapter 8, quite loosley.Putting a pair of linear equations into matrix notation and then solving by finding the inverse of the coefficient matrix. 

Choose which of 5 matrices are in a) row echelon form but not reduced b) reduced row echelon form c) neither.

Demo of automatically generating latex strings to out put vectors/matrices of variable size and that are calculated by some formula.

Matrix multiplication. Contains a function that will let you print the calculation steps of matrix multiplication, e.g. in the Advice.

simple sums of matrices and scalar mult of matrices.

Finding a matrix from a formula for each entry, which involves the row and column numbers of that entry. Not randomized because it's the same as in our workbook. But the variables are made in a way that it should be easy to randomise the size of the matrix, and the to change the formula for the input in not too many places.

Finding a matrix from a formula for each entry, which involves the row and column numbers of that entry. Randomized size of the matrices and formula.

Calculate matrix times vector.

Calculate matrix times vector.

checking by size whether two matrices can be multiplied. Student either gives size of resulting product, or NA if matrices can't be multiplied.

checking by size whether two matrices can be multiplied. Student either gives size of resulting product, or NA if matrices can't be multiplied.

First compute matrix times vector for specific vectors. Then determine domain and codomain and general formula for the matrix transformation defined by the matrix.

First compute matrix times vector for specific vectors. Then determine domain and codomain and general formula for the matrix transformation defined by the matrix.

Randomising the number of rows in the matrix, m, makes the marking algorithm for part c) slightly complicated: it checks whether it should include the third gap or not depending on the variable m. For the correct distribution of marks, it is then necessary to do "you only get the marks if all gaps are correct". Otherwise the student would only get 2/3 marks when m=2, so the third gap doesn't appear.

A combination of tasks: checking which matrix products exist, calculating some of these products, calculating transpose matrices. Comparing product of transpose with transpose of product. Experiencing associativity of matrix multiplication. Not much randomisation, only in which matrix product is computed as second option.

Comprehensive solution written out in Advice.

Student can choose one of all possible matrix products from the matrices given. Meant for voluntary extra practice. No extensive solutions: referred to other questions for this.

Matrix multiplication. Has automatically generated "unresolved" matrix product to write in the solution.

Use matrix multiplication to get an equation for \(k\) which is then to be solved.

To understand matrix multiplication in terms of linear combinations of column vectors.

Asking the student to create examples of two matrices which multiply to zero but are not themselves the zero matrix. Then getting the student to think about some features of these examples.