Question 73-79 of 216: Algebra (Section A: Common) | CUET (Common University Entrance Test) UG Mathematics Common (319) | Includes PYQs | Get Solutions with Detailed Explanations

Let a and b be two real numbers such that ${\displaystyle a>1,b>1,}$ If ${\displaystyle A=\left(\begin{array}{cc}a& 0\\ 0& b\end{array}\right),}$ then ${\displaystyle \underset{n⇾\mathrm{\infty }}{\lim }{A}^{-n}}$ is

Let ${\displaystyle A={\left[aij\right]}_{4\times 4}}$ such that ${\displaystyle aij=\{\begin{array}{c}2wheni=j\\ 0wheni\ne j\end{array},\text{then}}$

${\displaystyle \left\{\frac{\det \left(\mathrm{adj}\left(\mathrm{adj}A\right)\right)}{7}\right\}}$ is (where ${\displaystyle \{\bullet \}}$ represents fractional part function)

Let ${\displaystyle A=\left[\begin{array}{cc}0& \alpha \\ 0& 0\end{array}\right]\text{and}{(A+I)}^{50}-50A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\mathrm{.}}$ Then the value of ${\displaystyle a+b+c+d}$ is

If A and B are two non-zero square matrices of the same order such that the product ${\displaystyle AB=0,}$ then

If the system of equations ${\displaystyle x+ky+3z=0,3x+ky-2z=0,2x+3y-4z=0}$ has non-trivial solution, then ${\displaystyle \left(\frac{xy}{z^2}\right)=}$ …

${\displaystyle \left|\begin{array}{ccc}n& {}_{}{}^{n}P_n^{}& {}_{}{}^{n}C_n^{}\\ n+1& n+1_{P_{n+1}}& n+1_{C_{n+1}}\\ n+2& n+2_{P_{n+2}}& n+2_{C_{n+2}}\end{array}\right|=\mathrm{\dots }}$

Let ${\displaystyle a,b,c}$ be such that ${\displaystyle b(a+c)\ne 0}$ . If ${\displaystyle \left|\begin{array}{ccc}a& a+1& a-1\\ -b& b+1& b-1\\ c& c-1& c+1\end{array}\right|+\left|\begin{array}{ccc}a+1& b+1& c-1\\ a-1& b-1& c+1\\ {(-1)}^{n+2}a& {(-1)}^{n+1}b& {(-1)}^{n}c\end{array}\right|=0}$ then ${\displaystyle n}$ is