<p><strong>996.</strong> Let \(P\) be a \(2 \times 2\) matrix such that \(P\begin{bmatrix}1\\-1\end{bmatrix} = \begin{bmatrix}-1\\2\end{bmatrix}\) and \(P^2\begin{bmatrix}1\\-1\end{bmatrix} = \begin{bmatrix}1\\0\end{bmatrix}\). If \(x_1\) and \(x_2\) are two values of \(x\) for which \(|P - xI| = 0\), where \(I\) is an identity matrix of order 2, then find the value of \(x_1^2 + x_2^2\).</p>
Step-by-Step Solution
Key Concept: Use the given conditions to find relationships between matrix entries, then apply Cayley-Hamilton theorem or eigenvalue properties to find the sum of squares of eigenvalues using the trace and determinant of P.
<p><strong>Step 1: Extract information from the given conditions.</strong></p><p>Given: P[1, -1]ᵀ = [-1, 2]ᵀ and P²[1, -1]ᵀ = [1, 0]ᵀ</p><p>Let v = [1, -1]ᵀ. Then Pv = [-1, 2]ᵀ and P²v = [1, 0]ᵀ.</p><p><strong>Step 2: Apply P to both sides of the first equation.</strong></p><p>P(Pv) = P[-1, 2]ᵀ</p><p>P²v = P[-1, 2]ᵀ = [1, 0]ᵀ</p><p><strong>Step 3: Set up a system to find P applied to [-1, 2]ᵀ.</strong></p><p>Let P = [[a, b], [c, d]]. From Pv = [-1, 2]ᵀ:</p><p>a(1) + b(-1) = -1 ⟹ a - b = -1 ... (i)</p><p>c(1) + d(-1) = 2 ⟹ c - d = 2 ... (ii)</p><p><strong>Step 4: Use P²v = [1, 0]ᵀ and Pv = [-1, 2]ᵀ.</strong></p><p>P(Pv) = P[-1, 2]ᵀ = [1, 0]ᵀ</p><p>a(-1) + b(2) = 1 ⟹ -a + 2b = 1 ... (iii)</p><p>c(-1) + d(2) = 0 ⟹ -c + 2d = 0 ... (iv)</p><p><strong>Step 5: Solve for matrix entries.</strong></p><p>From (i) and (iii): a - b = -1 and -a + 2b = 1</p><p>Adding: b = 0, so a = -1</p><p>From (ii) and (iv): 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 6: Calculate trace and determinant of P.</strong></p><p>tr(P) = -1 + 2 = 1</p><p>det(P) = (-1)(2) - (0)(4) = -2</p><p><strong>Step 7: Find x₁² + x₂².</strong></p><p>The characteristic equation is |P - xI| = 0, giving eigenvalues x₁ and x₂.</p><p>By Vieta's formulas: x₁ + x₂ = tr(P) = 1 and x₁x₂ = det(P) = -2</p><p>x₁² + x₂² = (x₁ + x₂)² - 2x₁x₂ = (1)² - 2(-2) = 1 + 4 = 5</p><p><strong>∴ Answer: 5</strong></p>
Correct Answer: 5