M L Aggarwal Solutions for Chapter: Determinants, Exercise 1: EXERCISE 4.1

Find the value of $x$ if

$\left|\begin{array}{cc}x+2& 3\\ x+5& 4\end{array}\right|=3$.

Find the value of $x$ If 

$\left|\begin{array}{cc}3x& 7\\ -2& 4\end{array}\right|=\left|\begin{array}{cc}8& 7\\ 6& 4\end{array}\right|$.

If $\left|\begin{array}{cc}\sqrt{6}& x\\ \sqrt{20}& \sqrt{24}\end{array}\right|=\left|\begin{array}{cc}6& 2\\ 5& 2\end{array}\right|$, then write the positive value of $x^2$.

If $x\in \mathrm{I}$ and  $\left|\begin{array}{cc}2x& 3\\ -1& x\end{array}\right|=\left|\begin{array}{cc}3& 1\\ x& 3\end{array}\right|$, find the value of $x$.

If $\mathrm{A}=\left[\begin{array}{cc}1& 2\\ 4& 2\end{array}\right]$, then show that $\left|2\mathrm{A}\right|=4\left|\mathrm{A}\right|$.

Evaluate the following determinants:

$\left|\begin{array}{ccc}2& 4& 1\\ 8& 5& 2\\ -1& 3& 7\end{array}\right|$.

Evaluate the following determinant:

$\left|\begin{array}{ccc}3& -4& 5\\ 1& 1& -2\\ 2& 3& 1\end{array}\right|$

Prove that

$\left|\begin{array}{ccc}1& a& b\\ -a& 1& c\\ -b& -c& 1\end{array}\right|=1+a^2+b^2+c^2$.