Matrices and Determinants · Mathematics · TS EAMCET

If $x=k$ satisfies the equation $\left|\begin{array}{ccc}x-2 & 3 x-3 & 5 x-5 \\\\ x-4 & 3 x-9 & 5 x-25 \\\\ x-8 & 3 x-27 & 5 x-125\end{array}\right|=0$, then $x=k$ also satisfies the equation

If $A$ is a non-singular matrix, then $\operatorname{adj}\left(A^{-1}\right)=$

If the homogeneous system of linear equations $x-2 y+3 z=0,2 x+4 y-5 z=0,3 x+\lambda y+\mu z=0$ has non-trivial solution, then $8 \mu+11 \lambda=$

If $\frac{x^{2}}{2 x^{4}+7 x^{2}+6}=\frac{A x+B}{x^{2}+a}+\frac{C x+D}{a x^{2}+3}$, then $A+B+C-2 D=$

$A=\left[a_{i j}\right]$ is a $3 \times 3$ matrix with positive integers as its elements. Elements of $A$ are such that the sum of all elements of each row is equal to 6 and $a_{22}=2$.

If $\mathrm{a}_{i j}=\left\{\begin{array}{cl}\mathrm{a}_{i j}+\mathrm{a}_{j i}, & j=i+1 \text { when } i < 3 \\ \mathrm{a}_{i j}+\mathrm{a}_{j i}, & j=4-i \text { when } i=3\end{array}\right.$ for $i=1,2,3$, then $|\mathrm{A}|=$

If $|\operatorname{adj} A|=x$ and $|\operatorname{adj} B|=y$, then $\left|(\operatorname{adj}(A B))^{-1}\right|=$

The system of equations $x+3 b y+b z=0, x+2 a y+a z=0$ and $x+4 c y+c z=0$ has

$\left|\begin{array}{ccc}\frac{-b c}{a^{2}} & \frac{c}{a} & \frac{b}{a} \\ \frac{c}{b} & -\frac{a c}{b^{2}} & \frac{a}{b} \\ \frac{b}{c} & \frac{a}{c} & -\frac{a b}{c^{2}}\end{array}\right|=$

If $A=\left[\begin{array}{lll}x & y & y \\ y & x & y \\ y & y & x\end{array}\right]$ is a matrix such that $5 A^{-1}=\left[\begin{array}{ccc}-3 & 2 & 2 \\ 2 & -3 & 2 \\ 2 & 2 & -3\end{array}\right]$, then $A^2-4 A=$

If $A=\left[\begin{array}{lll}9 & 3 & 0 \\ 1 & 5 & 8 \\ 7 & 6 & 2\end{array}\right]$ and $A A^T-A^2=\left[\begin{array}{lll}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}\end{array}\right]$, then $\sum\limits_{\substack{1 \leq i \leq 3 \\ 1 \leq j \leq 3}} a_{i j}=$

If $a \neq b \neq c, \Delta_1=\left[\begin{array}{lll}1 & a^2 & b c \\ 1 & b^2 & c a \\ 1 & c^2 & a b\end{array}\right]$, $\Delta_2=\left[\begin{array}{ccc}1 & 1 & 1 \\ a^2 & b^2 & c^2 \\ a^3 & b^3 & c^3\end{array}\right]$ and $\frac{\Delta_1}{\Delta_2}=\frac{6}{11}$, then $11(a+b+c)=$

The system of equations $x+3 y+7=0$, $3 x+10 y-3 z+18=0$ and $3 y-9 z+2=0$ has

If $\alpha, \beta$ and $\gamma$ are the roots of the equation $\left|\begin{array}{lll}x & 2 & 2 \\ 2 & x & 2 \\ 2 & 2 & x\end{array}\right|=0$ and $\min (\alpha, \beta, \gamma)=\alpha$, then $2 \alpha+3 \beta+4 \gamma$ is equal to

If $\mathrm{A}=\left[\begin{array}{lll}1 & 2 & 2 \\ 3 & 2 & 3 \\ 1 & 1 & 2\end{array}\right]$ and $\mathrm{A}^{-1}=\left[\begin{array}{lll}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}\end{array}\right]$, then $\sum_{\substack{1 \leq i \leq 3 \\ 1 \leq j \leq 3}} a_{i j}=$

If $A X=D$ represents the system of linear equations $3 x-4 y+7 z+6=0,5 x+2 y-4 z+9=0$ and $8 x-6 y-z+5=0$, then

If $(x, y, z)=(\alpha, \beta, \gamma)$ is the unique solution of the system of simultaneous linear equations $3 x-4 y+z+7=0$, $2 x+3 y-z=10$ and $x-2 y-3 z=3$, then $\alpha=$

If $\alpha, \beta, \gamma$ are the roots of the equation $2 x^3-5 x^2+4 x-3=0$, then $\Sigma \alpha \beta(\alpha+\beta)=$