Matrices & Determinants
Homogeneous Linear Systems
Grade 12
Question:
<p>Given set, <i>A</i> = { <i>X</i> = (<i>x</i>, <i>y</i>, <i>z</i>)<sup>T</sup> : <i>PX</i> = 0 and <i>x</i><sup>2</sup> + <i>y</i><sup>2</sup> + <i>z</i><sup>2</sup> = 1, where <i>P</i> = \[\begin{pmatrix} 1 & 2 & 1 \\ -2 & 3 & -4 \\ 1 & 9 & -1 \end{pmatrix}\] }. Find the number of elements in set <i>A</i>.</p>
Step-by-Step Solution
Key Concept: A line through the origin intersects a unit sphere at exactly two diametrically opposite points. The homogeneous system with zero determinant generates a line through the origin.
<p><strong>Step 1:</strong> Calculate |<i>P</i>| = $$\begin{vmatrix} 1 & 2 & 1 \\ -2 & 3 & -4 \\ 1 & 9 & -1 \end{vmatrix}$$ = 1(−3 + 36) − 2(2 + 4) + 1(−18 − 3) = 33 − 12 − 21 = 0</p><p><strong>Step 2:</strong> Since |<i>P</i>| = 0, the homogeneous system <i>PX</i> = 0 has infinitely many solutions. According to the rank condition, since the rank is 2, the solution set forms a unique line passing through the origin.</p><p><strong>Step 3:</strong> The constraint <i>x</i><sup>2</sup> + <i>y</i><sup>2</sup> + <i>z</i><sup>2</sup> = 1 represents a sphere of radius 1 centered at the origin.</p><p><strong>Step 4:</strong> A line passing through the origin intersects a sphere of radius 1 at exactly two points (diametrically opposite).</p><p>∴ Set <i>A</i> contains exactly <strong>2 elements</strong>.</p>
Correct Answer: 2