Let $P_n$ denote the vector space over the reals of polynomials $p(x)$ of degree $n$  with coefficients in the real numbers.

Let the linear map $\phi: P_4 \rightarrow P_4$ be defined by:

$\phi(p(x))=ap(x) + (bx + c)p'(x) + (x ^ 2 + dx + f)p''(x)$

Using the standard basis for range and domain find the matrix given by $\phi$.

Given a matrix in row reduced form use this to find bases for the null, column and row spaces of the matrix.

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

Multiplication of $2 \times 2$ matrices.

Elementary Exercises in multiplying matrices. 

Multiplication of two matrices.

This question tests students knowledge of basic matrix arithmetic.

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

This question concerns the evaluation of the eigenvalues and corresponding eigenvectors of a 2x2 matrix.

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

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 $(x=1)$.

Cofactors Determinant and inverse of a 3x3 matrix.

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

Elementary Exercises in multiplying matrices. 

Multiplication of $2 \times 2$ matrices.

Cofactors Determinant and inverse of a 3x3 matrix.

Solving a system of three linear equations in 3 unknowns using Gauss Elimination in 4 stages. Solutions are all integral.

Elementary Exercises in multiplying matrices. 

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

(Last three parts of original question removed.)

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

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

rebelmaths

$A$ a $3 \times 3$ matrix. Using row operations on the augmented matrix $\left(A | I_3\right)$ reduce to $\left(I_3 | A^{-1}\right)$.

Elementary Exercises in multiplying matrices. 

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 $(x=1)$.

Cofactors Determinant and inverse of a 3x3 matrix.

Elementary Exercises in multiplying matrices. 

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

(Last three parts of original question removed.)

Cofactors Determinant and inverse of a 3x3 matrix.

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

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