Let $M$ be a $2 \times 2$ symmetric matrix with integer entries. Then $M$ is invertible, if:
The first column of $M$ is the transpose of the second row of $M$.
The second row of $M$ is the transpose of first column of $M$
$M$ is a diagonal matrix with non-zero entries in the main diagonal
The product of entries in the main diagonal of $M$ is not the square of an integer
Step-by-Step Solution
Key Concept: For a 2×2 symmetric matrix M = [[a, b], [b, c]], invertibility requires det(M) = ac - b² ≠ 0. Options 1-2 create linear dependence making det(M) = 0, while option 3 with non-zero diagonal entries in a diagonal matrix guarantees non-zero determinant, and option 4 ensures ac ≠ b².
Given matrix $M = \begin{bmatrix} a & b \\ b & c \end{bmatrix}$, we analyze the conditions in each option. For option (A), if $\begin{bmatrix} a \\ b \end{bmatrix}$ and $\begin{bmatrix} b \\ c \end{bmatrix}$ are transposes, then $a = b = c$. For option (B), if $\begin{bmatrix} b \\ c \end{bmatrix}$ and $\begin{bmatrix} a \\ b \end{bmatrix}$ are transposes, then $a = b = c$. For option (C), if $M = \begin{bmatrix} a & 0 \\ 0 & c \end{bmatrix}$, then $|M| = ac \neq 0$ requires $ac \neq 0$. Option (D) states that given $ac + \lambda^2$, we have a condition on eigenvalues.
Correct Answer: 3,4