Matrices & Determinants
Orthogonal matrix
Grade 12
Question:
<p>Which of the following is an orthogonal matrix?</p>
<p>\(\begin{bmatrix} 6/7 & 2/7 & -3/7 \\ 2/7 & 3/7 & 6/7 \\ 3/7 & -6/7 & 2/7 \end{bmatrix}\)</p>
<p>\(\begin{bmatrix} 6/7 & 2/7 & 3/7 \\ 2/7 & -3/7 & 6/7 \\ 3/7 & 6/7 & -2/7 \end{bmatrix}\)</p>
<p>\(\begin{bmatrix} -6/7 & -2/7 & -3/7 \\ 2/7 & 3/7 & 6/7 \\ -3/7 & 6/7 & 2/7 \end{bmatrix}\)</p>
<p>\(\begin{bmatrix} 6/7 & -2/7 & 3/7 \\ 2/7 & 2/7 & -3/7 \\ -6/7 & 2/7 & 3/7 \end{bmatrix}\)</p>
Step-by-Step Solution
Key Concept: A matrix is orthogonal if A·Aᵀ = Aᵀ·A = I, meaning its rows and columns are orthonormal vectors. Alternatively, |A| = ±1 and A⁻¹ = Aᵀ.
<p><strong>Step 1:</strong> For a matrix to be orthogonal, verify that A·A<sup>T</sup> = I (identity matrix).</p><p><strong>Step 2:</strong> Check that each row has unit length: sum of squares of elements in each row = 1.</p><p><strong>Step 3:</strong> Check that rows are mutually perpendicular: dot product of any two different rows = 0.</p><p><strong>Step 4:</strong> Verify det(A) = ±1 as a necessary condition.</p><p><strong>Step 5:</strong> The candidate matrix satisfying all orthonormality conditions is orthogonal.</p><p>∴ Answer: A</p>
Correct Answer: A