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+173−178
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C
173+178
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D
173+176−179
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Solution
The correct option is D173+176−179 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(A1∪A2)=P(A1)+P(A2)−P(A1∩A2)
(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