Embibe Experts Solutions for Chapter: Matrices and Determinants, Exercise 3: EXERCISE-3

If $D_r=\left|\begin{array}{ccc}{2}^{r-1}& 2\left(3^{r-1}\right)& 4\left(5^{r-1}\right)\\ x& y& z\\ 2^n-1& 3^n-1& 5^n-1\end{array}\right|$ then the value of $\sum _{r=1}^n{D}_r$ is

If $\left|\begin{array}{lll}(\beta +\gamma -\alpha -\delta {)}^4& (\beta +\gamma -\alpha -\delta {)}^2& 1\\ (\gamma +\alpha -\beta -\delta {)}^4& (\gamma +\alpha -\beta -\delta {)}^2& 1\\ (\alpha +\beta -\gamma -\delta {)}^4& (\alpha +\beta -\gamma -\delta {)}^2& 1\end{array}\right|=\mu \left(\alpha -\beta \right)\left(\alpha -\gamma \right)\left(\alpha -\delta \right)\left(\beta -\gamma \right)\left(\beta -\delta \right)\left(\gamma -\delta \right)$, then the value of $\mu$ is

The value of  $\left|\begin{array}{c}\begin{array}{ccc}{a}_1{l}_1+b_1 m_1& a_1{l}_2+b_1 m_2& a_1{l}_3+b_1 m_3\\ a_2{l}_1+b_2 m_1& a_2{l}_2+b_2 m_2& a_2{l}_3+b_2 m_3\\ a_3{l}_1+b_3 m_1& a_3{l}_2+b_3 m_2& a_3{l}_3+b_3 m_3\end{array}\end{array}\right|$ is equal to

If  $(a, b, c)$ is a solution of following sets of equations:

$x+y+z-6=0$,

$2x+y-z-1=0$,

$x+y-2z+3=0$ then the value of  $\left(\frac{c}{ab}\right)$ is

If $(a, b, c)$ is a solution of following sets of equations:

$x+2y+z=1$,

$3 x+y+z=6$,

$x+2y=0$ then the value of  $\left(\frac{bc}{a}\right)$ is

If  $\lambda \ne k$ for which system of equations $x+y+z=6; x+2y+3z=10$ & $x+2y+\lambda z=\mu$ have a unique solution, then the value of $k$ is

Let ${\alpha }_1, {\alpha }_2$ and ${\beta }_1, {\beta }_2$  be the roots of $a{x}^2+bx+c=0$ and $p{x}^2+qx+r=0$ respectively. If the system of equations ${\alpha }_{1}y+{\alpha }_{2}z=0$ and ${\beta }_{1}y+{\beta }_{2}z=0$ has a non-trivial solution and $\frac{b^2}{q^2}=\lambda \left(\frac{ac}{pr}\right)$, then value of $\frac{1}{\lambda }$ is

Let  $A=\left[\begin{array}{ll}a& b\\ c& d\end{array}\right]$ and  $P=\left[\begin{array}{l}p\\ q\end{array}\right]\ne \left[\begin{array}{l}0\\ 0\end{array}\right].$ Such that $\mathrm{AP}=P$ and  $a+d=5050$. Find the value of $(ad-bc)$.