Let An is a n×n Matrix in which diagonal elements are 1, 2, 3,…n (ie a11=1,a22=2,a33=3,……ann=n and all other elements are equal to n then

det ⋅ A n = 0 , det ⋅ A n = ( − 1 ) n + 1 n !

A n = 1 n n … n n 2 n … n n n 3 … n n n n … n ∴ A n is non-singular for all ' n ' A n = ( − 1 ) n + 1 ⋅ n !

det ⋅ A n = ( − 1 ) n + 1 n !

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Let $A_{n} is a n \times n$ Matrix in which diagonal elements are 1, 2, 3,…n $(ie a_{11} = 1, a_{22} = 2, a_{33} = 3, \dots \dots a_{nn} = n)$ and all other elements are equal to n then

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A_n is non-singular for all n

$\det \cdot A_{n} = 0$

$\det \cdot A_{n} = (- 1)^{n + 1} n!$

A_n is singular for all n

answer is B, C.

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$A_{n} = (\begin{array}{llll} 1 n n \dots n \\ n 2 n \dots n \\ n n 3 \dots n \\ n n n \dots n \end{array})$

$∴ A_{n} is non-singular for all ' n' |A_{n}| = (- 1)^{n + 1} \cdot n!$

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