Let M be a 2 x 2 symmetric matrix with integer entries. Then M is invertible if (a)The first column of M is the transpose of the second row of M (b)The second row of Mis the transpose of the first olumn 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 Mis not the square of an integer

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 digonal
D. the product of entries in the main diagonal of M is not the square of an integer

Explanation

PLAN A square matrix M is invertible, iff dem (M) or `|M| ne 0`
Let ` M = [{:(a, b), (b, c):}]`
(a) Given, `[(a), (b)] = [(b), (c)] rArr a =b =c =alpha " "["let"]`
`rArr M = [{:(alpha, alpha), (alpha, alpha):}] rArr |M| = 0 rArr M` is non-invertible.
(b) Given, [b c] = [a b]
`rArr a = b =c =alpha " "["let"]`
Again, |M| =0
`rArr` M is non-invertible.
(c) As given `M = [{:(a, 0), (0, c):}] rArr |M| = ac ne0`
[`because` a and c are non-zero]
`rArr ` M is invertible
(d) `M = [{:(a, b), (b,c):}] rArr |M|= ac-b^(2) ne 0`
`because` ac is not equal to square of an integer.
M is invertible.