Matrices & Determinants
Matrix Inverse and Properties
Grade 12
Question:
<p>Let <m>P = \begin{pmatrix} 1 & 0 & 0 \\ 3 & 1 & 0 \\ 9 & 3 & 1 \end{pmatrix}</m> and <m>Q = [q_{ij}]</m> be two 3×3 matrices such that <m>Q - P^{-1} = I_3</m>. Then, <m>\frac{5(q_{21} + q_{31})}{q_{32}}</m> is equal to</p>
<p>(a) 10</p>
<p>(b) 135</p>
<p>(c) 9</p>
<p>(d) 15</p>
Step-by-Step Solution
Key Concept: Find the inverse of a lower triangular matrix, then use the relation Q = P⁻¹ + I to identify matrix elements.
<p><strong>Step 1:</strong> Find <m>P^{-1}</m> from the lower triangular matrix <m>P</m>.</p><p><strong>Step 2:</strong> Calculate <m>Q = P^{-1} + I_3</m>.</p><p><strong>Step 3:</strong> Extract elements <m>q_{21}, q_{31}, q_{32}</m> and compute the given expression.</p><p>∴ Answer is (d) 15.</p>
Correct Answer: D