![Given matrices A = [[k, 4], [3, 2]] and B = [[1, 2], [0, 4]], find the value of k such that the determinant of the product AB is 4.](/_next/image?url=https%3A%2F%2Fohijasnjvprzodcmxltt.supabase.co%2Fstorage%2Fv1%2Fobject%2Fpublic%2Fwhatsapp-images%2F1776970372336-f334224a20f9426a.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: Find the product of matrices A and B. Given matrices: A = k & 4 \\ 3 & 2 and B = 1 & 2 \\ 0 & 4 Multiply A by B: AB = k & 4 \\ 3 & 2 1 & 2 \\ 0 & 4 AB = (k)(1) + (4)(0) & (k)(2) + (4)(4) \\ (3)(1) + (2)(0) & (3)(2) + (2)(4) AB = k + 0 & 2k + 16 \\ 3 + 0 & 6 + 8 AB = k & 2k + 16 \\ 3 & 14 Step 2: Calculate the determinant of the product matrix AB. For a 2 × 2 matrix a & b \\ c & d , the determinant is ad - bc. (AB) = (k)(14) - (2k + 16)(3) (AB) = 14k - (6k + 48) (AB) = 14k - 6k - 48 (AB) = 8k - 48 Step 3: Set the determinant equal to 4 and solve for k. We are given that (AB) = 4. 8k - 48 = 4 Add 48 to both sides of the equation: 8k = 4 + 48 8k = 52 Divide by 8: k = (52)/(8) Simplify the fraction: k = (13)/(2) k = 6.5 The value of k is 6.5. Send me the next one 📸