Let M be a `2xx2` symmetric matrix with integer entries.
Then , M is invertible, if

Options

A. the first column of M is the transpose of the second row of
M
B. The second row of M is the transpose of the first column of
M
C. M is a diagonal matrix with non- zero entries in the main
diagonal
D. the product of entries in the main diagonal of M is not the
square of an integer

Answer

C, D

Explanation

Let `M= [[a,b],[c,d]]`, where `a, b, c, in I`
M is invertible if `abs((a,b),(b,c)) ne 0 rArr ac- b^(2) ne 0 `
(a) `[[a],[b]]=[[b],[c]]rArr a = b =c rArr ac-b^(2)=0`
`therefore` Option (a) is incorrect
(b) `[(b,c)]= [(a,b)] rArr a = b = c rArr ac - b^(2) = 0`
`therefore` Option (b) is incorrect
(c) `M= [[a,0],[0,c]], ` then` abs(M) = ac ne 0`
`therefore` M is invertible
`therefore` Potion ( c) is correct.
(d) As `acne"Integre """^(2)rArrac ne b^(2)`
`therefore ` Option (d)is correct.