![Given matrix P = [[2, 1], [-3, 0]], calculate P2 - 2P + 4I.](/_next/image?url=https%3A%2F%2Fohijasnjvprzodcmxltt.supabase.co%2Fstorage%2Fv1%2Fobject%2Fpublic%2Fwhatsapp-images%2F1776467522722-71f8d34cd8f86f32.png&w=3840&q=75)
This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Step 1: Identify the given matrices. The matrix P is given as: P = 2 & 1 \\ -3 & 0 The 2 × 2 unit matrix I is: I = 1 & 0 \\ 0 & 1 Step 2: Calculate P^2. P^2 = P × P = 2 & 1 \\ -3 & 0 2 & 1 \\ -3 & 0 Perform matrix multiplication: P^2 = (2)(2) + (1)(-3) & (2)(1) + (1)(0) \\ (-3)(2) + (0)(-3) & (-3)(1) + (0)(0) P^2 = 4 - 3 & 2 + 0 \\ -6 + 0 & -3 + 0 P^2 = 1 & 2 \\ -6 & -3 Step 3: Calculate 2P. Multiply each element of matrix P by 2: 2P = 2 2 & 1 \\ -3 & 0 = 2 × 2 & 2 × 1 \\ 2 × (-3) & 2 × 0 2P = 4 & 2 \\ -6 & 0 Step 4: Calculate 4I. Multiply each element of matrix I by 4: 4I = 4 1 & 0 \\ 0 & 1 = 4 × 1 & 4 × 0 \\ 4 × 0 & 4 × 1 4I = 4 & 0 \\ 0 & 4 Step 5: Evaluate P^2 - 2P + 4I. Substitute the calculated matrices into the expression: P^2 - 2P + 4I = 1 & 2 \\ -6 & -3 - 4 & 2 \\ -6 & 0 + 4 & 0 \\ 0 & 4 First, perform the subtraction: 1 & 2 \\ -6 & -3 - 4 & 2 \\ -6 & 0 = 1 - 4 & 2 - 2 \\ -6 - (-6) & -3 - 0 = -3 & 0 \\ 0 & -3 Now, perform the addition: -3 & 0 \\ 0 & -3 + 4 & 0 \\ 0 & 4 = -3 + 4 & 0 + 0 \\ 0 + 0 & -3 + 4 = 1 & 0 \\ 0 & 1 The result is the 2 × 2 unit matrix. Comparing this to the given options, it matches option C. The final answer is 1 & 0 \\ 0 & 1 . Send me the next one 📸