Matrices & Determinants
Linear Transformations
Grade 12
Question:
<p>The matrix <span class="math">\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}</span> is the matrix reflection in the line</p>
<p>(a) <span class="math">x = 1</span></p>
<p>(b) <span class="math">x + y = 1</span></p>
<p>(c) <span class="math">y = 1</span></p>
Step-by-Step Solution
Key Concept: A matrix that swaps coordinates represents reflection across the line y = x or its equivalent form x - y = 0.
<p>The matrix <span class="math">\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}</span> represents a reflection. Testing by applying it to a point: <span class="math">\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} y \\ x \end{pmatrix}</span>. This swaps <span class="math">x</span> and <span class="math">y</span>, which is reflection in the line <span class="math">y = x</span> (or equivalently <span class="math">x - y = 0</span>).</p>
Correct Answer: b