Subject Experts Solutions for Chapter: Determinants, Exercise 1: Practice Exercise

If the value of the determinant $\left|\begin{array}{lll}a& 1& 1\\ 1& b& 1\\ 1& 1& c\end{array}\right|$ is positive, where $a\ne b\ne c,$ then the value of $abc$

If $a,b$ and $c$ are real numbers, then the value of the determinant $\left|\begin{array}{lll}1-a& a-b-c& b+c\\ 1-b& b-c-a& c+a\\ 1-c& c-a-b& a+b\end{array}\right|$ is

Consider the following statements in respect of the determinant $\left|\begin{array}{cc}{\cos }^2\frac{\alpha }{2}& {\sin }^2\frac{\alpha }{2}\\ {\sin }^2\frac{\beta }{2}& {\cos }^2\frac{\beta }{2}\end{array}\right|$ where $\alpha ,\beta$ are complementary angles.

$\mathrm{I}$. The value of the determinant is $\frac{1}{\sqrt{2}}\cos \left(\frac{\alpha -\beta }{2}\right)$.

$\mathrm{II}$. The maximum value of the determinant is $\frac{1}{\sqrt{2}}$.

Which of the above statement $\left(s\right)$ is $/$ are correct$?$

The matrix $A=\left[\begin{array}{ccc}1& 3& 2\\ 1& x-1& 1\\ 2& 7& x-3\end{array}\right]$ will have inverse for every real number $x$ except for

If $A$ is an orthogonal matrix of order $3$ and $B=\left[\begin{array}{ccc}1& 2& 3\\ -3& 0& 2\\ 2& 5& 0\end{array}\right]$, then which of the following is/are correct $?$

$\mathrm{I}$. $\left|AB\right|=\pm 47$

$\mathrm{II}$. $AB=BA$

Select the correct answer using the code given below.

If $A$ is a square matrix, then what is $adj\left(A^{-1}\right)-{\left(adjA\right)}^{-1}$ equal to?

Which of the following determinants have value zero?

$\mathrm{I}$. $\left|\begin{array}{lll}41& 1& 5\\ 79& 7& 9\\ 29& 5& 3\end{array}\right|$

$\mathrm{II}$. $\left|\begin{array}{lll}1& a& b+c\\ 1& b& c+a\\ 1& c& a+b\end{array}\right|$

$\mathrm{III}$.$\left|\begin{array}{rrr}0& c& b\\ -c& 0& a\\ -b& -a& 0\end{array}\right|$

Select the correct answer using the code given below. 

The system of linear equations $kx+y+z=1$, $x+ky+z=1$ and $x+y+kz=1$ has a unique solution under which one of the following conditions?