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Question

# If An is a n×n matrix in which the diagonal elements are 1,2,3.....n (i.e., a11=1a22=2,.......aii=i,......ann=n) and all other elements are equal to n, then

A
An is singular for all n
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B
An is non singular for all n
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C
det An = (1)nn!
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D
det An = (1)n+1n!
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Solution

## The correct options are B An is non singular for all n D det An = (−1)n+1n!For n=2,A2=[1222]|A2|=−2≠0⇒|A2|=(−1)2+12!For n=3,A3=⎡⎢⎣133323333⎤⎥⎦|A3|=6≠0⇒|A3|=(−1)3+13!For n=4, A4=⎡⎢ ⎢ ⎢⎣1444424444344444⎤⎥ ⎥ ⎥⎦|A4|=−24≠0⇒|A4|=(−1)4+14!Hence, An is non-singular for all n.and |An|=(−1)n+1n!

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