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Question

Matrices of order 3×3 are formed using the elements of set A={3,2,1,0,1,2,3}. Then the probability that matrices are either symmetric or skew-symmetric, is

A
176+173
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B
176+173178
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C
173+178
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D
173+176179
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Solution

The correct option is D 173+176179
A={3,2,1,0,1,2,3}
Let P(A1) be the probability that the matrix is symmetric.
and P(A2) be the probability that the matrix is skew-symmetric.
P(A1A2)=P(A1)+P(A2)P(A1A2)

(a) P(A1)
For symmetric matrix, AT=A
A=777177117
(Each entry denotes the number of ways with which it can be filled)
Favourable cases =76
Total cases =79
P(A1)=7679=173

(b) P(A2)
For skew-symmetric matrix, AT=A
A=077107110
(Each entry denotes the number of ways with which it can be filled)
Favourable cases =73
Total cases =79
P(A2)=7379=176

(c) P(A1A2) (Null matrix)
=179

Thus, P(A1A2)=173+176179

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