If A and B are n times n diagonalizable matrices , then A+B is also diagonalizable. True or False?

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.
$\text{Basis }=\{\left[\begin{array}{cc}& \\ & \end{array}\right],\left[\begin{array}{cc}& \\ & \end{array}\right]\}$

Let W be the subspace of all diagonal matrices in $M_{2,2}$. Find a bais for W. Then give the dimension of W.
If you need to enter a matrix as part of your answer , write each row as a vector.For example , write the matrix

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

Determine whether the statement, $(A+B{)}^{-1}=A^{-1}+B^{-1}$, assuming A, B, and A + B are invertible, is true or false. If the statement is false, make the necessary change(s) to produce a true statement.

Verify if the statement is true: The eigenvalues of a linear transformation T in L(U) are independent of the chosen base of U.

Show that B is the inverse of A.
$A=\left[\begin{array}{cc}1& -1\\ -1& 2\end{array}\right],B=\left[\begin{array}{cc}2& 1\\ 1& 1\end{array}\right]$