Let A = $\left[\begin{array}{cc}2& -1\\ 1& 1\end{array}\right]$ . If the sum of the diagonal elements of A¹³ is 3ⁿ , then n is equal to ______.

If $\alpha$$\ne$a, $\beta \ne$b, $\gamma \ne$ b, $\gamma \ne$c, and $\left|\begin{array}{ccc}\alpha & b& c\\ a& \beta & c\\ a& b& \gamma \end{array}\right| = 0$ then

$\frac{a}{\alpha - a} + \frac{b}{\beta - b} + \frac{\gamma }{\gamma - c}$ is equal to:

Let $\lambda , \mu \in R$

If the system of equations
3x + 5y + $\lambda$z = 3
7x + 11y –9z = 2
97x + 155y – 189z = $\mu$
has infinitely many solutions, then $\mu$ +2$\lambda$ is equal
to :

Let B = $\left[\begin{array}{cc}1& 3\\ 1& 5\end{array}\right]$ and A be a 2 × 2 matrix such that $A{B}^{–1} = A^{–1} . If BC{B}^{–1}$

= A and $C^4 + \alpha C^2 + \beta I =\mathit{0}, then 2\beta - \alpha$ is equal to :

Consider the matrices : A = $\left[\begin{array}{cc}2& -5\\ 3& \mathrm{m}\end{array}\right], \mathrm{B}= \left[\begin{array}{c}20\\ \mathrm{m}\end{array}\right]$ and X = $\left[\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right]$. Let the set of all m, for which the

system of equations AX = B has a negative solution

(i.e., x < 0 and y < 0), be the interval (a, b).

Then $8{\int }_{\mathrm{a}}^{\mathrm{b}} \left|\mathrm{A}\right| dm$ is equal to ________.

For a $3 \times 3$ matrix M, let trace (M) denote the sum of all the diagonal elements of M. Let A be a $3 \times 3$ matrix such that $\left|\mathrm{A}\right|=\frac{1}{2}$ and trace $\left(\mathrm{A}\right) = 3$. If $\mathrm{B} = \mathrm{adj}\left(\mathrm{adj}\right(2\mathrm{A}\left)\right)$, then the value of $\left|\mathrm{B}\right| + \mathrm{trace} \left(\mathrm{B}\right)$ equals: