Let `M` be a `2xx2` 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 `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

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 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,c),(c,b)]" "("where "a, b, c, in I)`
(1) If the first column of M is the transpose of the second row of M, then
`[(a,c)]=[(c,b)]`
`:. A=b=c`
Thus, det. `(M) =ab-c^(2)=0`
Hence, `M` is not invertible.
(2) If the second row of `M` is the transpose of the first column of M, then
`[(c, b)]=[(a, c)]`
`:. a=b=c`
thus, det `(M)=ab-c^(2)=0`
Hence, `M` is not invertible.
(3) If `M=[(a,0),(0,b)]`, with `a, b, ne 0`, then
det. `(M)=ab ne 0`
Hence, `M` is invertible.
(4) If product of elements in main diagonal which `(ab)` is not perfect square, then
det. `(M)=ab-c^(2) ne 0`
Hence, `M` is invertiable.