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 $(x=1)$.

Calculate the dot products of (a) two 2D vectors and (b) two 3D vectors.

Given $6$ vectors in $\mathbb{R^4}$ and given that they span $\mathbb{R^4}$ find a  basis.

Given the following three vectors $\textbf{v}_1,\;\textbf{v}_2,\;\textbf{v}_3$ Find out whether they are a linearly independent set are not. Also if linearly dependent find the relationship $\textbf{v}_{r}=p\textbf{v}_{s}+q\textbf{v}_{t}$ for suitable $r,\;s,\;t$ and integers $p,\;q$.

Given $5$ vectors in $\mathbb{R^4}$ determine if a spanning set for $\mathbb{R^4}$ or not by looking for any simple dependencies between the vectors.

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

Given a 3x3 matrix with very big elements, perform row operations to find a matrix with single-digit elements. Then reduce that to an upper triangular matrix, and hence find the determinant.

Find the determinant of  a random (3x3) matrix.

Calculate an answer involving a fractional index.

Calculate an answer involving a fractional index.

Using indices rules to write powers as negative powers

Writing negative powers as fractions

Writing fractins as negative powers

Using indices rules to write expressions as negative powers.

Using indices rules to write fractions as negative powers.

Calculate an answer involving a fractional index.

Calculate an answer involving a fractional index.

Calculate an answer involving a fractional index.

Using rules for roots and powers to write expressions with only powers

Using indices to write roots as fractional powers

Using indices rules to rewrite an expression from $a^\frac{m}{n}$ to $b$, for integers $a$, $b$, $m$ and $n$.

Calculate an answer involving a fractional index.

Simplifying expressions from $(z^m)^n$ to $z^{mn}$. 

Simplifying expressions from $y^m\div y^n$ to $y^{m-n}$. 

Simplifying expressions from $x^mx^n$ to $x^{m+n}$. 

Simplifying expressions from $(8^m)^n$ to $8^{mn}$. 

Simplifying expressions from $3^m\div 3^n$ to $5^{m-n}$ where $n$ can be bigger then $m$.

Simplifying expressions from $5^m\div 5^n$ to $5^{m-n}$. 

Simplifying expressions from $2^m2^n$ to $2^{m+n}$. 

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