Show that B is the multiplicative inverse of A, where: A=begin{bmatrix}2 & 1 1 & 1 end{bmatrix} text{ and } B=begin{bmatrix}1 & -1 -1 & 2 end{bmatrix}

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 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.

Let A,B,C be $n\times n$ matrices. If $B=CA{C}^{-1}$ show that $det(A)=det(B)$

Write the matrix equation as a system of linear equations without matrices.
$\left[\begin{array}{ccc}2& 0& -1\\ 0& 3& 0\\ 1& 1& 0\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}6\\ 9\\ 5\end{array}\right]$

Use elementary matrices to write ? in row reduced echelon form. Use the elementary matrices to find the inverse of
$A=\left[\begin{array}{ccc}1& 0& -2\\ 0& 2& 1\\ 0& 0& 1\end{array}\right]$

Determine if the system has a nontrivial solution. Try to use as few row operations as possible.
${\displaystyle 2x_1-5x_2+8x_3=0}$
${\displaystyle -2x_1-7x_2+x_3=0}$
${\displaystyle 4x_1+2x_2+7x_3=0}$