Q77 / 160 · TS EAMCET 2023 (14 May Shift 2) · Mathematics

If a matrix is chosen at random from the set of all $3 \times 3$ non zero matrices whose entries are the elements of the set $\{-1,0,1\}$, then the probability that the matrix is skew symmetric is

  • A
    $\frac{1}{729}$
  • Correct answer
    B
    $\frac{1}{757}$
  • C
    $\frac{1}{703}$
  • D
    $\frac{1}{742}$

Solution

Total numbers of non-zero matrices $=3^9-1$ Now to make skew symmetric, diagonal elements must be 0 $\left[\begin{array}{ccc}0 & a & b \\ -a & 0 & c \\ -b & -c & 0\end{array}\right]$ also if we take $a$ from $-1,0$ or 1 then $-a$ is also fixed, similarly for $b$ Hence, total number of skew symmetric matrices $=3^3$ Out of which if we select $a=b=c=0$ then it will be zero matrix $\therefore$ Non zero skew symmetric matrices $=3^3-1$ Probability $=\frac{3^3-1}{3^9-1}=\frac{26}{19682}=\frac{1}{757}$

Hence, the correct answer is option: 'B'.

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