Matrices & Determinants
Orthogonal Matrices
Grade 12
Question:
<p>If matrix <span class="math">A</span> is orthogonal with rows <span class="math">R_1, R_2, R_3</span>, and <span class="math">R_1 \times R_3 = 0</span> gives <span class="math">x + 4 + 2y = 0</span>, and <span class="math">R_2 \times R_3 = 0</span> gives <span class="math">2x + 2 - 2y = 0</span>, find <span class="math">xy</span>.</p>
Step-by-Step Solution
Key Concept: Orthogonal matrix rows are mutually perpendicular, so their dot products equal zero. Use this to set up and solve a system of linear equations.
<p><strong>Solution:</strong></p><p>Since <span class="math">A</span> is orthogonal, each row is orthogonal to other rows:</p><p>From <span class="math">R_1 \times R_3 = 0</span>: <span class="math">x + 4 + 2y = 0</span></p><p>From <span class="math">R_2 \times R_3 = 0</span>: <span class="math">2x + 2 - 2y = 0</span></p><p>Solving these simultaneous equations:</p><p>From the second equation: <span class="math">x + 1 - y = 0 \Rightarrow x = y - 1</span></p><p>Substituting into the first: <span class="math">(y-1) + 4 + 2y = 0 \Rightarrow 3y + 3 = 0 \Rightarrow y = -1</span></p><p>Therefore: <span class="math">x = -2</span>, <span class="math">y = -1</span></p><p><span class="math">\therefore xy = (-2)(-1) = 2</span></p>
Correct Answer: 2