Question 12.2. Using row operations, find the inverse of
$\begin{pmatrix} 1 & 1 & -1 \ 0 & 0 & 1 \ 2 & 1 & 2 \end{pmatrix}$
Question 12.3. Consider the matrix $A = \begin{pmatrix} -1 & 0 \ 4 & 5 \end{pmatrix}$,
(i) Find elementary matrices $E_1, E_2, E_3$ such that $E_3E_2E_1A = I_2$,
(ii) Find the 2 \times 2 matrix $B$ satisfying the following relation:
$(I_2 + 2B^T)^{-1} = A$.
Question 12.4. Suppose that $D$ is a 3 \times 3 diagonal matrix with diagonal entries
$\frac{1}{2}, \frac{\lambda}{3}, \lambda$, where $\lambda \neq 0$. If $tr(D^{-1}) = 3$, find the value of $\lambda$.
Question 12.5. Let $A$ and $B$ be tow square matrices of the same size. If $A$ is
symmetric and $B$ is skew-symmetric, show that $AB - BA$ is symmetric.
Question 12.6. Find the values of $a$ and $b$ so that $x_1 = 3$ and $x_2 = 2$ is a solution
of the following linear system:
$ax_1 + (2b + 1)x_2 = 1$
$(a + b)x_1 - 4x_2 = -8.$
