Determine the null space of each of the following matrices: begin{pmatrix}1 & 3 &-4 2 & -1 & -1 -1 & -3 &4 end{pmatrix}

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.

Determine if b is a linear combination of the vectors formed from the columns of the matrix A.
$A=\left[\begin{array}{ccc}1& -4& 2\\ 0& 3& 5\\ -2& 8& -4\end{array}\right],b=\left[\begin{array}{c}3\\ -7\\ -3\end{array}\right]$

A drug production company has three branches A, B, and C, each of which produces two types of medicine daily. If the first branch produces 100 boxes of the first type, 400 boxes of the second type, and the branch B produces 200 boxes of the first type, 300 boxes of the type The second and branch C produce 150 boxes of the first type, 250 boxes of the second type.
A) Systems the data in a matrix in two ways
B) Are the two matrices equal? Why?

Find values for the variables so that the matrices are equal. $\left[\begin{array}{c}x\\ 4\end{array}\right]=\left[\begin{array}{c}6\\ y\end{array}\right]$

$\left(\begin{array}{cc}1& 2\\ 3& -2\end{array}\right)+\left(\begin{array}{cc}-6& -5\\ -2& -2\end{array}\right)=$ ?

Matrix multiplication is pretty tough- so i will cover that in class. In the meantime , compute the following if
$A=\left[\begin{array}{ccc}2& 1& 1\\ -1& -1& 4\end{array}\right],B=\left[\begin{array}{cc}0& 2\\ -4& 1\\ 2& -3\end{array}\right],C=\left[\begin{array}{cc}6& -1\\ 3& 0\\ -2& 5\end{array}\right],D=\left[\begin{array}{ccc}2& -3& 4\\ -3& 1& -2\end{array}\right]$
If the operation is not possible , write NOT POSSIBLE and be able to explain why
a)A+B
b)B+C
c)2A