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
To solve this problem, we need to interpret the given notation. The expression "If x + iy " implies that the complex number x+iy is equal to the determinant of the given matrix. Step 1: Calculate the determinant of the matrix. Let the given matrix be A: A = 6i & -3i & 1 \\ 4 & 3i & -1 \\ 20 & 3 & i The determinant of a 3 × 3 matrix a & b & c \\ d & e & f \\ g & h & i is given by a(ei - fh) - b(di - fg) + c(dh - eg). Applying this formula to our matrix: (A) = 6i | 3i & -1 \\ 3 & i | - (-3i) | 4 & -1 \\ 20 & i | + 1 | 4 & 3i \\ 20 & 3 | Step 2: Calculate the 2 × 2 determinants. 1. The first minor: | 3i & -1 \\ 3 & i | = (3i)(i) - (-1)(3) = 3i^2 + 3 Since i^2 = -1: 3(-1) + 3 = -3 + 3 = 0 2. The second minor: | 4 & -1 \\ 20 & i | = (4)(i) - (-1)(20) = 4i + 20 3. The third minor: | 4 & 3i \\ 20 & 3 | = (4)(3) - (3i)(20) = 12 - 60i Step 3: Substitute the minor determinants back into the main determinant formula. x + iy = 6i(0) - (-3i)(4i + 20) + 1(12 - 60i) x + iy = 0 + 3i(4i + 20) + 12 - 60i Distribute 3i: x + iy = 12i^2 + 60i + 12 - 60i Substitute i^2 = -1: x + iy = 12(-1) + 60i + 12 - 60i x + iy = -12 + 60i + 12 - 60i Combine the real and imaginary parts: x + iy = (-12 + 12) + (60i - 60i) x + iy = 0 + 0i x + iy = 0 Step 4: Determine x and y, then find x - iy. From x + iy = 0, we can equate the real and imaginary parts: x = 0 y = 0 Now, we need to find x - iy: x - iy = 0 - i(0) x - iy = 0 Comparing this result with the given options: a. 0 b. 1 + 3i c. 3i d. 3 + i The calculated value matches option a. The final answer is a. 0.