Find the matrices: a)A + B b) A - B c) -4A d)3A + 2B. A=begin{bmatrix}3&1 &1-1&2&5 end{bmatrix} , B=begin{bmatrix}2&-3 &6-3&1&-4 end{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.
$\text{Basis }=\{\left[\begin{array}{cc}& \\ & \end{array}\right],\left[\begin{array}{cc}& \\ & \end{array}\right]\}$

For the following Leslie matrix , find an approximate expression for the population distribution after n years , given that the initial population distribution is given by $X(0)=\left[\begin{array}{c}2000\\ 4000\end{array}\right],L^n=\left[\begin{array}{cc}0.8& 0.4\\ 1.2& 0\end{array}\right]$
Select the correct choice below and fill in the answer boxes to complete your choise.
a)$X\approx ()({)}^n\left[\begin{array}{c}1\\ ()\end{array}\right]$
b)$X\approx ({)}^n\left[\begin{array}{c}1\\ ()\end{array}\right]$

Use the given of the coefficient matrix to solve the following system.
${\displaystyle 7x_1+3x_2=6}$
${\displaystyle -6x_1-3x_2=4}$
$$A^{-1}=\left[\begin{array}{cc}1& 1\\ -2& -\frac{7}{3}\end{array}\right]$$

find the product of AB?
$A=\left[\begin{array}{cc}3& 1\\ 6& 0\\ 5& 0\end{array}\right]$
$B=\left[\begin{array}{ccc}1& 5& 3\\ -1& 2& -3\end{array}\right]$
a) $\left[\begin{array}{ccc}2& 17& 6\\ 6& 30& 18\\ 5& 25& 15\end{array}\right]$
b) $\left[\begin{array}{ccc}16& 6& 18\\ -2& 10& -30\\ 25& 17& 6\end{array}\right]$
c) $\left[\begin{array}{ccc}21& 6& -23\\ 2& 30& 5\\ -25& 12& 17\end{array}\right]$
d) The product is not defined

Determine whether the subset of $M_{n,n}$ is a subspace of $M_{n,n}$ with the standard operations. Justify your answer.
The set of all $n\times n$ matrices whose entries sum to zero

Prove that if B is symmetric, then $\Vert B\Vert$ is the largest eigenvalue of B.