Amit M Agarwal Solutions for Chapter: Determinant, Exercise 3: Target Exercises

If $a>b>c$ and the system of equations $ax+by+cz=0$, $bx+cy+az=0$ and $cx+ay+bz=0$ has a non-trivial solution, then both the roots of the quadratic equation $a{t}^2+bt+c=0$ are

The multiplicative inverse of $A=\left[\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right]$ is

If $a, b$ and $c$ are non-zero real numbers, then the inverse of the matrix $A=\left[\begin{array}{lll}a& 0& 0\\ 0& b& 0\\ 0& 0& c\end{array}\right]$ is

The multiplicative inverse of the matrix $A=\left[\begin{array}{lll}0& 0& 1\\ 0& 1& 0\\ 1& 0& 0\end{array}\right]$ is

The value of third order determinant is $11,$ then the value of the square of the determinant formed by the cofactors will be

If a system of the equations ${\left(\alpha +1\right)}^{3}x+{\left(\alpha +2\right)}^{3}y-{\left(\alpha +3\right)}^3=0, \left(\alpha +1\right)x+\left(\alpha +2\right)y-\left(\alpha +3\right)=0$ and $x+y-1=0$ is consistent. Then, what is the value of $\alpha$

The rank of the matrix $\left[\begin{array}{llll}4& 1& 0& 0\\ 3& 0& 1& 0\\ 6& 0& 2& 0\end{array}\right]$ is

If $f\left(x\right)=\left|\begin{array}{ccc}x& \sin x& \cos x\\ x^2& \tan x& -x^3\\ 2x& \sin 2x& 5x\end{array}\right|,$ then $\underset{x\to 0}{\lim }\frac{f^{\text{'}}\left(x\right)}{x}$ equals