<p>If <strong>A</strong> is a 3 × 3 matrix and <strong>u</strong> is a vector. If <strong>Au</strong> and <strong>u</strong> are orthogonal for all real <strong>u</strong>, then matrix <strong>A</strong> is a</p>
Step-by-Step Solution
Key Concept: If two vectors are orthogonal, their dot product is zero. For <strong>Au</strong> ⋅ <strong>u</strong> = 0 to hold for all non-zero <strong>u</strong>, the matrix must be singular.
<p><strong>Given:</strong> <strong>Au</strong> ⋅ <strong>u</strong> = 0</p><p><strong>Step 1:</strong> Since <strong>Au</strong> and <strong>u</strong> are orthogonal, we have <strong>Au</strong> ⋅ <strong>u</strong> = 0</p><p><strong>Step 2:</strong> This means <strong>u</strong>\(^T\) <strong>A</strong>\(^T\) <strong>u</strong> = 0</p><p><strong>Step 3:</strong> For this to be true for all non-zero <strong>u</strong>, we must have |<strong>A</strong>| = 0</p><p><strong>∴ Answer is (a):</strong> <strong>A</strong> is singular.</p>
Correct Answer: a