Matrices & Determinants
Matrix Multiplication and Properties
Grade 12
Question:
<p>If <m>A = \begin{pmatrix} 1 & 2 & 2 \\ 2 & 1 & -2 \\ 2 & -2 & -1 \end{pmatrix}</m> is a matrix satisfying the equation <m>AA^T = 9I</m>, where <m>I</m> is 3×3 identity matrix, then <m>(a, b)</m> is equal to</p>
<p>(a) (2, -1)</p>
<p>(b) (-2, 1)</p>
<p>(c) (2, 1)</p>
<p>(d) (-2, -1)</p>
Step-by-Step Solution
Key Concept: Use the condition AAᵀ = 9I to create equations that allow solving for the unknown parameters in the matrix.
<p><strong>Step 1:</strong> Calculate <m>A^T</m> (transpose of A).</p><p><strong>Step 2:</strong> Compute <m>AA^T</m> and equate to <m>9I</m>.</p><p><strong>Step 3:</strong> From the equation <m>AA^T = 9I</m>, identify the unknown elements <m>a</m> and <m>b</m>.</p><p>∴ Answer is (a) (2, -1).</p>
Correct Answer: A