![Given that A = [2 4; 3 6] and B = [11 3; 4 1] find C such that BC = A. 2. Find the inverse of the matrix [2 5; 3 4]. Hence or otherwise solve the equations 2x + 5y = 9 and 3x + 4y = 6.](/_next/image?url=https%3A%2F%2Fohijasnjvprzodcmxltt.supabase.co%2Fstorage%2Fv1%2Fobject%2Fpublic%2Fwhatsapp-images%2F1782206077752-5fb7319dc33df126.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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here are the solutions to the matrix problems: 1. Given that A = 2 & 4 \\ 3 & 6 and B = 11 & 3 \\ 4 & 1 , find C such that B · C = A. Step 1: To find C, we need to multiply both sides of the equation B · C = A by the inverse of B (B^-1) from the left. This gives C = B^-1 · A. First, calculate the determinant of B. (B) = (11)(1) - (3)(4) = 11 - 12 = -1 Step 2: Calculate the inverse of B. B^-1 = (1)/((B)) 1 & -3 \\ -4 & 11 = (1)/(-1) 1 & -3 \\ -4 & 11 = -1 & 3 \\ 4 & -11 Step 3: Multiply B^-1 by A to find C. C = B^-1 · A = -1 & 3 \\ 4 & -11 2 & 4 \\ 3 & 6 C = (-1)(2) + (3)(3) & (-1)(4) + (3)(6) \\ (4)(2) + (-11)(3) & (4)(4) + (-11)(6) C = -2 + 9 & -4 + 18 \\ 8 - 33 & 16 - 66 C = 7 & 14 \\ -25 & -50 The matrix C is 7 & 14 \\ -25 & -50 . 2. Find the inverse of the matrix 2 & 5 \\ 3 & 4 . Hence or otherwise solve the equations 2x + 5y = 9 and 3x + 4y = 6. Part 1: Find the inverse of the matrix. Step 1: Let the given matrix be M = 2 & 5 \\ 3 & 4 . First, calculate the determinant of M. (M) = (2)(4) - (5)(3) = 8 - 15 = -7 Step 2: Calculate the inverse of M. M^-1 = (1)/((M)) 4 & -5 \\ -3 & 2 = (1)/(-7) 4 & -5 \\ -3 & 2 = -(4)/(7) & (5)/(7) \\ (3)/(7) & -(2)/(7) The inverse of the matrix is -(4)/(7) & (5)/(7) \\ (3)/(7) & -(2)/(7) . Part 2: Solve the equations using the inverse matrix. Step 1: Write the system of equations in matrix form. 2 & 5 \\ 3 & 4 x \\ y = 9 \\ 6 Step 2: Multiply both sides by the inverse matrix M^-1 found in Part 1. x \\ y = M^-1 9 \\ 6 = -(4)/(7) & (5)/(7) \\ (3)/(7) & -(2)/(7) 9 \\ 6 Step 3: Perform the matrix multiplication. x \\ y = (-(4)/(7))(9) + ((5)/(7))(6) \\ ((3)/(7))(9) + (-(2)/(7))(6) x \\ y = -(36)/(7) + (30)/(7) \\ (27)/(7) - (12)/(7) x \\ y = -(6)/(7) \\ (15)/(7) The solution to the equations is x = -(6)/(7) and y = (15)/(7). What's next? Send 'em.