Question

If a set $A$ is $\{1,0,-1,2\}$ then which of the following is(are) correct?

• A. A square matrix of order 3 is constructed whose elements are from set $A$ then the probability that matrix is skew symmetric is $\frac{27}{4^9}$.
• B. A square matrix of order 3 is constructed whose elements are from set A then the probability that matrix is symmetric is $\frac{1}{64}$.
• C. A function $f:A\to A$ is defined then probability that function is onto is $\frac{3}{32}$.
• D. A function $f:A\to A$ is defined and $f$ is onto then the probability that $f(i)\ne i,\forall i\in A$ is $\frac{3}{8}$.

Answer 1

Answer: (A), (B), (C), (D)

$\Theta A=\{1,0,-1,2\}$
(a) For skew symmetric matrix, all diagonal elements can be filled in only one way by zero.
For $a_{ij}=-a_{ji}$
$\therefore$ For every $a_{ij}(i\ne j)$ there are three ways, then for corresponding $a_{ji}$ there is only way
$\therefore$ Required probability $=\frac{3\times 3\times 3}{4^9}=\frac{27}{4^9}$
(b) For matrix to be symmetric, $a_{ij}=a_{ji}$
$\therefore$ Probability $=\frac{4\times 4\times 4\times 4\times 4\times 4\times 1\times 1\times 1}{4^9}=\frac{1}{64}$
(c) Total number of function $=4^4$
number of onto functions $=4$ !
$\therefore$ Probability $=\frac{4!}{4^4}=\frac{24}{256}=\frac{3}{32}$
(d) $\Theta f(i)\ne i$
$\therefore$ number of function $=4!\left(\frac{1}{2!}-\frac{1}{3!}+\frac{1}{4!}\right)=9$
$\therefore$ Probability $=\frac{9}{4!}=\frac{9}{24}=\frac{3}{8}$