Question

Let $S$ be the set of all $2\times 2$ symmetric matrices whose entries are either zero or one. A matrix $X$ is chosen from $S$. The probability that the determinant of $X$ is not zero is

• A. $\frac{1}{3}$
• B. $\frac{1}{2}$
• C. $\frac{3}{4}$
• D. $\frac{1}{4}$
• E. $\frac{2}{9}$

Answer 1

Answer: (B)

$S=\{2\times 2$ symmetric matrices whose entries are either zero or one $\}$
$\left\{\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right]\left[\begin{array}{cc}0& 0\\ 0& 1\end{array}\right]\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]\left[\begin{array}{cc}0& 1\\ 1& 1\end{array}\right]\left[\begin{array}{cc}1& 1\\ 1& 0\end{array}\right]\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right]\right\}$
$\therefore n(s)=8$
Let $x=\{$ matrix whose determinant is non-zero $\}$
$\left\{\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]\left[\begin{array}{cc}0& 1\\ 1& 1\end{array}\right]\left[\begin{array}{cc}1& 1\\ 1& 0\end{array}\right]\right\}$
$\therefore n(x)=4$
$\therefore P(x)=\frac{n(X)}{n(S)}$
$=\frac{4}{8}=\frac{1}{2}$