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 matrix A and the function f(x). The matrix A is given as: A = 1 & 2 \\ -1 & 0 The function f(x) is given as: f(x) = x^2 - 1 To find f(A), we need to substitute A into the function, which means calculating A^2 - I, where I is the identity matrix of the same dimension as A. For a 2 × 2 matrix, the identity matrix is I = 1 & 0 \\ 0 & 1 . Step 2: Calculate A^2. A^2 = A × A = 1 & 2 \\ -1 & 0 1 & 2 \\ -1 & 0 Perform matrix multiplication: (A^2)_11 = (1)(1) + (2)(-1) = 1 - 2 = -1 (A^2)_12 = (1)(2) + (2)(0) = 2 + 0 = 2 (A^2)_21 = (-1)(1) + (0)(-1) = -1 + 0 = -1 (A^2)_22 = (-1)(2) + (0)(0) = -2 + 0 = -2 So, A^2 is: A^2 = -1 & 2 \\ -1 & -2 Step 3: Calculate f(A) = A^2 - I. f(A) = -1 & 2 \\ -1 & -2 - 1 & 0 \\ 0 & 1 Perform matrix subtraction: f(A)_11 = -1 - 1 = -2 f(A)_12 = 2 - 0 = 2 f(A)_21 = -1 - 0 = -1 f(A)_22 = -2 - 1 = -3 Thus, f(A) is: f(A) = -2 & 2 \\ -1 & -3 Step 4: Compare the result with the given options. The calculated matrix f(A) = -2 & 2 \\ -1 & -3 matches option B. The final answer is B. -2 & 2 \\ -1 & -3 .