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  • A second-order determinant is written down at random using the numbers 1,-1 as elements. The probability that the value of the determinant is nonzero isOpti

A second-order determinant is written down at random using the numbers 1,-1 as elements. The probability that the value of the determinant is nonzero is

Option: 1

\frac{1}{2}


Option: 2

\frac{3}{8}


Option: 3

\frac{5}{8}


Option: 4

\frac{1}{3}


Answers (1)

best_answer

\mathrm{n(S)=2 \times 2 \times 2 \times 2=16}

because each of the four places can be filled in 2 ways. 

The zero determinants are \left|\begin{array}{rr} 1 & 1 \\ -1 & -1 \end{array}\right|,\left|\begin{array}{rr} -1 & -1 \\ 1 & 1 \end{array}\right|,\left|\begin{array}{ll} 1 & 1 \\ 1 & 1 \end{array}\right|,\left|\begin{array}{ll} -1 & -1 \\ -1 & -1 \end{array}\right|, \left|\begin{array}{ll} -1 & 1 \\ -1 & 1 \end{array}\right|,\left|\begin{array}{ll} 1 & -1 \\ 1 & -1 \end{array}\right|,\left|\begin{array}{rr} -1 & 1 \\ 1 & -1 \end{array}\right|,\left|\begin{array}{rr} 1 & -1 \\ -1 & 1 \end{array}\right| .

\mathrm{\therefore \quad P(E)=\frac{8}{16}=\frac{1}{2} .}

Posted by

HARSH KANKARIA

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