<p>Let <span>\(P = \begin{bmatrix} a & b \\ c & d \end{bmatrix}\)</span> be a <span>\(2 \times 2\)</span> matrix such that <span>\(P\begin{bmatrix}1\\-1\end{bmatrix} = \begin{bmatrix}-1\\2\end{bmatrix}\)</span> and <span>\(P \cdot P\begin{bmatrix}1\\-1\end{bmatrix} = \begin{bmatrix}1\\0\end{bmatrix}\)</span>. If <span>\(x_1, x_2\)</span> are the eigenvalues of <span>\(P\)</span>, find <span>\(x_1^2 + x_2^2\)</span>.</p>
Step-by-Step Solution
Key Concept: Use the two matrix equations to find P, then compute trace(P²) - 2·det(P) = λ₁² + λ₂² using the Cayley-Hamilton identity and eigenvalue properties.
<p><strong>Step 1:</strong> From the first condition, P[1, -1]ᵀ = [-1, 2]ᵀ, we get:</p><p>a - b = -1 and c - d = 2</p><p><strong>Step 2:</strong> From the second condition, P·P[1, -1]ᵀ = [1, 0]ᵀ, which means P[-1, 2]ᵀ = [1, 0]ᵀ:</p><p>-a + 2b = 1 and -c + 2d = 0</p><p><strong>Step 3:</strong> Solve the system:</p><p>From equations (1) and (3): a - b = -1 and -a + 2b = 1</p><p>Adding: b = 0, so a = -1</p><p>From equations (2) and (4): c - d = 2 and -c + 2d = 0</p><p>Adding: d = 2, so c = 4</p><p>Therefore, P = [[-1, 0], [4, 2]]</p><p><strong>Step 4:</strong> Calculate trace(P) = -1 + 2 = 1 and det(P) = (-1)(2) - (0)(4) = -2</p><p><strong>Step 5:</strong> For eigenvalues: x₁² + x₂² = (x₁ + x₂)² - 2x₁x₂ = [trace(P)]² - 2·det(P)</p><p>= (1)² - 2(-2) = 1 + 4 = 5</p><p>∴ Answer: <strong>5</strong></p>
Correct Answer: 5