<p>Let \(M\) be a \(2 \times 2\) symmetric matrix with integer entries. Then \(M\) is invertible if</p>
<p>the first column of \(M\) is the transpose of the second row of \(M\)</p>
<p>the second row of \(M\) is the transpose of the first column of \(M\)</p>
<p>\(M\) is a diagonal matrix with non-zero entries in the main diagonal</p>
<p>the product of entries in the main diagonal of \(M\) is not the square of an integer</p>
Step-by-Step Solution
Key Concept: A symmetric matrix M is invertible if and only if det(M) ≠ 0. For a 2×2 symmetric matrix with integer entries, you must check when ad - b² ≠ 0 (where M = [[a,b],[b,d]]).
<p><strong>Step 1:</strong> Let M be a 2×2 symmetric matrix: M = [[a, b], [b, d]] where a, b, d are integers.</p><p><strong>Step 2:</strong> A matrix is invertible ⟺ det(M) ≠ 0.</p><p><strong>Step 3:</strong> Calculate det(M) = ad - b·b = ad - b².</p><p><strong>Step 4:</strong> Therefore, M is invertible if and only if <strong>ad - b² ≠ 0</strong>.</p><p><strong>Step 5:</strong> This means the discriminant of the quadratic form must be non-zero, or equivalently, the eigenvalues cannot both be zero.</p><p>∴ Answer: M is invertible if det(M) = ad - b² ≠ 0</p>
Correct Answer: CD