Matrices practice · Medium

Let $a_1, a_2, a_3, \ldots, a_{10}$ be in GP. with $a_i>0$ for $i=1,2, \ldots, 10$ and $\mathrm{S}$ be the set of pairs (r, k), $r, k \in N$ (the set of natural numbers) for which

$\left|\begin{array}{llll} \log _e \mathrm{a}_1^{\mathrm{r}} \mathrm{a}_2^{\mathrm{k}} & \log _e \mathrm{a}_2^{\mathrm{r}} \mathrm{a}_3^{\mathrm{k}} & \log _e \mathrm{a}_3^{\mathrm{r}} \mathrm{a}_4^{\mathrm{k}} \\ \log _e \mathrm{a}_4^{\mathrm{r}} \mathrm{a}_5^{\mathrm{k}} & \log _e \mathrm{a}_5^{\mathrm{r}} \mathrm{a}_6^{\mathrm{k}} & \log _e \mathrm{a}_6^{\mathrm{r}} \mathrm{a}_7^{\mathrm{k}} \\ \log _e \mathrm{a}_7^{\mathrm{r}} \mathrm{a}_8^{\mathrm{k}} & \log _e \mathrm{a}_8^{\mathrm{r}} \mathrm{a}_9^{\mathrm{k}} & \log _e \mathrm{a}_9^{\mathrm{r}} \mathrm{a}_{10}^{\mathrm{k}} \end{array}\right|=0$
Then the number of elements in S, is :

If $\mathrm{AB}=0$, then for the matrices $A = \left[ {\begin{array}{cccccccccccccccccccc} {{{\cos }^2}\theta }&{\cos \theta \sin \theta } \\ {\cos \theta \sin \theta }&{{{\sin }^2}\theta } \end{array}} \right]{\text{and}}\,\,B = \left[ {\begin{array}{cccccccccccccccccccc} {{{\cos }^2}\phi }&{\cos \phi \sin \phi } \\ {\cos \phi \sin \phi }&{{{\sin }^2}\phi } \end{array}} \right],\theta - \phi$ is

If $A=\left[\begin{array}{cc}-4 & -1 \\ 3 & 1\end{array}\right]$, then the determinant of the matrix $\left(A^{2016}-2 A^{2015}-A^{2014}\right)$ is :

If the system of equation $x+y+z=16 ; 2 x+5 y+\alpha z=\beta ; x+2 y+3 z=14$ has infinitely many solution, then $\alpha+\beta=$

The matrix of the transformation reflection in the line $x+y=0$ is

If $f\left( x \right) = \left| {\begin{array}{cccccccccccccccccccc} {x - 3}&{2{x^2} - 18}&{3{x^3} - 81}\\ {x - 5}&{2{x^2} - 50}&{4{x^3} - 500}\\ 1&2&3 \end{array}} \right|,$ then $f\left( 1 \right).f\left( 3 \right) + f\left( 3 \right).f\left( 5 \right) + f\left( 5 \right).f\left( 1 \right)$ is equal to

In $\Delta ABC$ if $\left| {\begin{array}{cccccccccccccccccccc} 1&a&b\\ 1&c&a\\ 1&b&c \end{array}} \right| = 0,$ then ${\sin ^2}A + {\sin ^2}B + {\sin ^2}C$ is equal to

Let $a, b, c$ be positive and not all equal, the value of the determinant $\left|\begin{array}{lll}a & b & c \\ b & c & a \\ c & a & b\end{array}\right|,$ is

If $a,b,c$ are unequal what is the condition that the value of the determinant, $\mathrm{\Delta } \equiv \left| {\begin{array}{cccccccccccccccccccc} a&{{a^2}}&{{a^3} + 1}\\ b&{{b^2}}&{{b^3} + 1}\\ c&{{c^2}}&{{c^3} + 1} \end{array}} \right|$ is 0?

If $A, B$ and $C$ denote the angles of a triangle, then $\Delta=\left|\begin{array}{ccc}-1 & \cos C & \cos B \\ \cos C & -1 & \cos A \\ \cos B & \cos A & -2\end{array}\right|$ is independent of

If $\left[\begin{array}{cc}1 / 25 & 0 \\ x & 1 / 25\end{array}\right]=\left\{\left[\begin{array}{cc}5 & 0 \\ -a & 5\end{array}\right]^{-1}\right\}^2$, then the value of $x$ is

If $\mathrm{A}=\left[\begin{array}{cc}1 & 1 \\ 0 & 1\end{array}\right]$ and $\mathrm{B}=\left[\begin{array}{cc}\frac{\sqrt{3}}{2} & \frac{1}{2} \\ \frac{-1}{2} & \frac{\sqrt{3}}{2}\end{array}\right]$, then $\left(B B^{\top} A\right)^5$ is equal to :

If $\mathrm{A}=\left[\begin{array}{cc}\sin \theta & -\cos \theta \\ \cos \theta & \sin \theta\end{array}\right]$, then $\mathrm{A}^{-1}=$

If $\mathrm{k} \left[\begin{array}{ccc}-1 & 2 & 2 \\ 2 & -1 & 2 \\ 2 & 2 & -1\end{array}\right]$ is an orthogonal matrix then $\mathrm{k}$ is equal to-

Statement 1: If the system of equations $x+k y+3 z=0$, $3 x+k y-2 z=0,2 x+3 y-4 z=0$ has a non-trivial solution, then the value of $k$ is $\frac{31}{2}$.

Statement 2: A system of three homogeneous equations in three variables has a non trivial solution if the determinant of the coefficient matrix is zero.

If $P=\left[\begin{array}{cc}\frac{\sqrt{3}}{2} & \frac{1}{2} \\ -\frac{1}{2} & \frac{\sqrt{3}}{2}\end{array}\right]$ and $A=\left[\begin{array}{ll}1 & 1 \\ 0 & 1\end{array}\right]$ and $Q=$ $P A P^T$ and $x=P^T Q^{2005} P$ then $x$ is equal to

If $\alpha, \beta, \gamma$ are different from 1 and are the roots of $a x^3+b x^2+c x+d=0$ and $(\beta-\gamma)(\gamma-\alpha)(\alpha-\beta)=25 / 2$, then the determinant $\Delta=\left|\begin{array}{ccc}\frac{\alpha}{1-\alpha} & \frac{\beta}{1-\beta} & \frac{\gamma}{1-\gamma} \\ \alpha & \beta & \gamma \\ \alpha^2 & \beta^2 & \gamma^2\end{array}\right|$ equals

If $l_r^2+m_r^2+n_r^2=1$, where $r=1,2,3$ and $l_1 l_2+m_1 m_2+n_1 n_2=0\ldots$ etc, then, $\Delta=\left|\begin{array}{lll}l_1 & m_1 & n_1 \\ l_2 & m_2 & n_2 \\ l_3 & m_3 & n_3\end{array}\right|^2$ is equal to