Let A,B be two matrices with linearly independent columns . If Col(A)=Col(B) then A(A^TA)^{-1}A^T=B(B^TB)^{-1}B^T True or False?

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.

$A=\left[\begin{array}{cc}5& 1\\ 1& 1\end{array}\right]$
$B=\left[\begin{array}{cc}3& 5\\ 1& 3\end{array}\right]$
Similar or not similar?

Determine which matrices are in reduced echelon form and which others are only in echelon form.
a)$\left[\begin{array}{cccc}1& 0& 1& 0\\ 0& 1& 1& 0\\ 0& 0& 0& 1\end{array}\right]$
b)$\left[\begin{array}{ccccc}0& 1& 1& 1& 1\\ 0& 0& 1& 1& 1\\ 0& 0& 0& 0& 1\\ 0& 0& 0& 0& 0\end{array}\right]$
c)$\left[\begin{array}{cccc}1& 5& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 1\end{array}\right]$ ф

Show that a square matrix which has a row or a column consisting entirely of zeros must be singular.

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

-x-+y=1 -3x+y=7What is the solution