Matrices & Determinants
Null Space
Grade 12
Question:
<p>Let \(A = \{X = (x, y, z)^T : PX = 0 \text{ and } x^2 + y^2 + z^2 = 1\}\), where \[P = \begin{pmatrix}1 & 2 & 1\\-2 & 3 & -4\\1 & 9 & -1\end{pmatrix}\]</p><p>Then the set <em>A</em></p>
<p>(a) is a singleton</p>
<p>(b) is an empty set</p>
<p>(c) contains more than two elements</p>
<p>(d) contains exactly two elements</p>
Step-by-Step Solution
Key Concept: The null space of a 3×3 matrix with certain rank properties intersects the unit sphere at exactly two points (antipodal points).
<p><strong>Analysis:</strong> Find the null space of matrix <em>P</em> by solving $PX = 0$. The null space will be a line through the origin (1-dimensional). The constraint $x^2 + y^2 + z^2 = 1$ represents a unit sphere. The intersection of a line through the origin with a unit sphere gives exactly two points (opposite points on the sphere).</p>
Correct Answer: D