Multiplication of $2 \times 2$ matrices.

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 $\phi$.

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: \[\phi(p(x))=p(a)+p(bx+c).\]Using the standard basis for range and domain find the matrix given by $\phi$.

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

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

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

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. 

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

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

Elementary Exercises in multiplying matrices. 

aij  notation and definition of the order of a matrix.

Questions on matrix arithmetic.

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Find eigenvalues and eigenvectors of a $2 \times 2$ matrix.

Find the determinants of three $2 \times 2$ invertible matrices.

Find the inverses of three $2 \times 2$ invertible matrices.

Three questions on linear combinations and products of matrices.

Solve a pair of linear equations by writing an equivalent matrix equation.

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

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: \[\phi(p(x))=p(a)+p(bx+c).\]Using the standard basis for range and domain find the matrix given by $\phi$.

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.