![Given that matrix K = [[8, 9x], [1, x]], find the (i) value of x for which the determinant of K is 2, (ii) inverse of K.](/_next/image?url=https%3A%2F%2Fohijasnjvprzodcmxltt.supabase.co%2Fstorage%2Fv1%2Fobject%2Fpublic%2Fwhatsapp-images%2F1783629653748-28b54787ddfe4083.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
Given the matrix K = 8 & 9x \\ 1 & x . a) (i) Find the value of x for which the determinant of K is 2. Step 1: Calculate the determinant of matrix K. For a 2 × 2 matrix a & b \\ c & d , the determinant is ad - bc. (K) = (8)(x) - (9x)(1) (K) = 8x - 9x (K) = -x Step 2: Set the determinant equal to 2 and solve for x. -x = 2 x = -2 The value of x is -2. a) (ii) Find the inverse of K. Step 1: Substitute the value of x = -2 into matrix K. K = 8 & 9(-2) \\ 1 & -2 K = 8 & -18 \\ 1 & -2 Step 2: Calculate the determinant of K with x = -2. From part (i), we know (K) = -x. (K) = -(-2) = 2 Alternatively, using the matrix with x=-2: (K) = (8)(-2) - (-18)(1) (K) = -16 - (-18) (K) = -16 + 18 = 2 Step 3: Find the inverse of K. For a 2 × 2 matrix a & b \\ c & d , the inverse is (1)/((K)) d & -b \\ -c & a . K^-1 = (1)/(2) -2 & -(-18) \\ -1 & 8 K^-1 = (1)/(2) -2 & 18 \\ -1 & 8 Step 4: Multiply each element by (1)/(2). K^-1 = (-2)/(2) & (18)/(2) \\ (-1)/(2) & (8)/(2) K^-1 = -1 & 9 \\ -(1)/(2) & 4 Send me the next one 📸