![Find the inverse of matrix A and solve the system of linear equations using matrices. A = [[3, 4], [2, 3]]](/_next/image?url=https%3A%2F%2Fohijasnjvprzodcmxltt.supabase.co%2Fstorage%2Fv1%2Fobject%2Fpublic%2Fwhatsapp-images%2F1777308149661-0101a26cdd38d7f6.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.
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You're on a roll — a) Given matrix A = 3 & 4 \\ 2 & 3 . To find the inverse of a 2 × 2 matrix A = a & b \\ c & d , the formula is A^-1 = (1)/((A)) d & -b \\ -c & a . Step 1: Calculate the determinant of A. (A) = (3)(3) - (4)(2) = 9 - 8 = 1 Step 2: Apply the inverse formula. A^-1 = (1)/(1) 3 & -4 \\ -2 & 3 = 3 & -4 \\ -2 & 3 The inverse matrix is 3 & -4 \\ -2 & 3 . b) Let x be the price of one bag of beans and y be the price of one bag of rice. From the given information, we can form a system of linear equations: For School A: 60x + 80y = 240000 For School B: 150x + 225y = 637500 Step 1: Write the system of equations in matrix form M x \\ y = C. 60 & 80 \\ 150 & 225 x \\ y = 240000 \\ 637500 Step 2: Find the determinant of the coefficient matrix M = 60 & 80 \\ 150 & 225 . (M) = (60)(225) - (80)(150) = 13500 - 12000 = 1500 Step 3: Find the inverse of the coefficient matrix M^-1. M^-1 = (1)/(1500) 225 & -80 \\ -150 & 60 Step 4: Multiply M^-1 by the constant matrix C to find x \\ y . x \\ y = (1)/(1500) 225 & -80 \\ -150 & 60 240000 \\ 637500 x \\ y = (1)/(1500) (225)(240000) + (-80)(637500) \\ (-150)(240000) + (60)(637500) x \\ y = (1)/(1500) 54000000 - 51000000 \\ -36000000 + 38250000 x \\ y = (1)/(1500) 3000000 \\ 2250000 x = (3000000)/(1500) = 2000 y = (2250000)/(1500) = 1500 The price of one bag of beans is Kshs. 2000 and the price of one bag of rice is Kshs. 1500. c) Step 1: Calculate the new price of beans. The price of beans decreased in the ratio 4:5. New price of beans = (4)/(5) × 2000 = 4 × 400 = 1600 Kshs. Step 2: Calculate the new price of rice. The price of rice increased by 20%. New price of rice = 1500 × (1 + 0.20) = 1500 × 1.20 = 1800 Kshs. Step 3: Calculate the total amount paid by the businessman. The businessman bought 20 bags of beans and 30 bags of rice. Cost of beans = 20 × 1600 = 32000 Kshs. Cost of rice = 30 × 1800 = 54000 Kshs. Total amount paid = 32000 + 54000 = 86000 Kshs. The businessman paid Kshs. 86,000. Last free one today — make it count tomorrow, or type /upgrade for unlimited.