The reduced row echelon form of the augmented matrix of a system of linear equations is given. Determine whether this system of linear equations is consistent and, if so, find its general solution. Write the solution in vector form. begin{bmatrix}1&-2&0&0&-30&0&1&0&-40&0&0&1&5end{bmatrix}

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

What are S I X fractions equivalent to 3/4?

Construct a ${\displaystyle 3\times 3}$ matrix A, with nonzero entries, and a vector b in ${\displaystyle {\mathbb{R}}^3}$ such that b is not in the set spanned by the columns of A.

write B as a linear combination of the other matrices, if possible.
$B=\left[\begin{array}{cc}2& 3\\ -4& 2\end{array}\right],A_1=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right],A_2=\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right],A_3=\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right]$

Describe all solutions of Ax=0 in parametric vector form, where A is row equivalent to the given matrix.
$$\left[\begin{array}{cccc}1& 3& 0& -4\\ 2& 6& 0& -8\end{array}\right]$$

Show that the product of two $2\times 2$ skew symmetric matrices is diagonal. Is this true for $n\times n$ skew symmetric matrices with n > 2?