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Question

(b) A determinant of order $2$ is of the form $\Delta = \left| {\,\begin{array}{*{20}{c}}a&b\c&d\end{array}\,} ight|$

It is equal to $ad

Vedclass · Accepted Answer

$\frac{3}{8}$

Vedclass · Answer

(b) A determinant of order $2$ is of the form $\Delta = \left| {\,\begin{array}{*{20}{c}}a&b\c&d\end{array}\,} ight|$

It is equal to $ad - bc.$

The total number of ways of choosing $a,\,\,b,\,\,c$ and $d$ is $2 	imes 2 	imes 2 	imes 2 = 16.$

Now $\Delta 
e 0$ if and only if either $ad = 1,$ $bc = 0$ or $ad = 0,$ $bc = 1.$ But $ad = 1,$ $bc = 0$ if $a = d = 1$ and one of $b,\,c$ is zero.

Therefore $ad = 1,$ $bc = 0$ in three cases,

similarly $ad = 0,$ $bc = 1$ in three cases.

Therefore the required probability $ = \frac{6}{{16}} = \frac{3}{8}.$

A determinant is chosen at random from the set of all determinants of order $2$ with elements $0$ or $1$ only. The probability that the determinant chosen is non-zero is

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