Let U=begin{bmatrix}frac{sqrt2}{2} & frac{sqrt3}{3} -frac{sqrt2}{2} & frac{sqrt3}{3} 0&-frac{sqrt3}{3} end{bmatrix} text{ and } x=begin{bmatrix}1 2 end{bmatrix} b)Compute | Ux | (it should be the same as | x |)

Find a basis for the space of $2\times 2$ diagonal matrices.
$\text{Basis }=\{\left[\begin{array}{cc}& \\ & \end{array}\right],\left[\begin{array}{cc}& \\ & \end{array}\right]\}$

Let B be a 4x4 matrix to which we apply the following operations:
1. double column 1,
2. halve row 3,
3. add row 3 to row 1,
4. interchange columns 1 and 4,
5. subtract row 2 from each of the other rows,
6. replace column 4 by column 3,
7. delete column 1 (column dimension is reduced by 1).
(a) Write the result as a product of eight matrices.
(b) Write it again as a product of ABC (same B) of three matrices.

Find the determinant of the matrix:
${\displaystyle \left[\begin{array}{ccc}2& 1& 4\\ -3& 1& 5\\ 4& 0& 0\end{array}\right]}$

Let $M\in R^{10\times 10},s.t.M^{2020}=0$. Prove $M^{10}=0$

Given matrix A and matrix B. Find (if possible) the matrices: (a) AB (b) BA.
$A=\left[\begin{array}{ccc}1& -1& 1\\ 5& 0& -2\\ 3& -2& 2\end{array}\right],B=\left[\begin{array}{ccc}1& 1& 0\\ 1& -4& 5\\ 3& -1& 2\end{array}\right]$

Determine whether A is similar B .If $A\sim B$, give an invertible matrix P such that $P^{-1}AP=B$
$A=\left[\begin{array}{ccc}1& 1& 0\\ 0& 1& 1\\ 0& 0& 1\end{array}\right],B=\left[\begin{array}{ccc}1& 1& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$

Subject: matrices
Determine:
$\left[\begin{array}{ccc}1& -1& 2\end{array}\right]\{\left[\begin{array}{c}4\\ -1\\ 0\end{array}\right]\left[\begin{array}{cc}2& 3\end{array}\right]+\left[\begin{array}{cc}1& 0\\ 0& 1\\ -1& -1\end{array}\right]\}$