Matrices of order `3xx3` are formed by using the elements of the set `A={-3,-2,-1,0,1,2,3}`, then probability that matrix is either symmetric or skew

Question

Matrices of order `3xx3` are formed by using the elements of the set `A={-3,-2,-1,0,1,2,3}`, then probability that matrix is either symmetric or skew

Matrices of order `3xx3` are formed by using the elements of the set `A={-3,-2,-1,0,1,2,3}`, then probability that matrix is either symmetric or skew symmetric is
A. `(1)/(7^(6))+(1)/(7^(3))`
B. `(1)/(7^(9))+(1)/(7^(3))-(1)/(7^(6))`
C. `(1)/(7^(3))+(1)/(7^(9))`
D. `(1)/(7^(3))+(1)/(7^(6))-(1)/(7^(9))`

Monjur Alahi

5 views

2025-01-04 04:42

1 Answers

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Salim Ahmed Kawsar · Accepted Answer · 2023-02-05T15:29:48+06:00

Correct Answer - D
`(d)` For symmetric matrix each place in upper triangle and leading diagonal can be filled in `7` ways. Then number of symmetric matrices are `7^(6)`.
For skew symmetric matrix leading diagonal elements are zero. Upper triangle elements can be filled in `7^(3)` ways.
The required probability is `=(7^(6))/(7^(9))+(7^(3))/(7^(9))-(1)/(7^(9))` (as one matric is common in both the cases)