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 $f : A \rightarrow A$ is defined then probability that function is onto is $\frac{3}{32}$.
• D. A function $f : A \rightarrow A$ is defined and $f$ is onto then the probability that $f ( i ) \neq i , \forall i \in A$ 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 $a_{i j}(i \neq j)$ there are three ways, then for corresponding $a_{j i}$ 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) \neq 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} $