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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Hey Stars, good to see you again. Here are the solutions to your matrix problems. Question 1: Given the matrix Q = 3 & 2 \\ x & 4 . Find the value of x given that the determinant of Q is 2. Step 1: Calculate the determinant of Q. For a 2 × 2 matrix a & b \\ c & d , the determinant is ad - bc. (Q) = (3)(4) - (2)(x) = 12 - 2x Step 2: Set the determinant equal to 2 and solve for x. 12 - 2x = 2 10 = 2x x = (10)/(2) x = 5 The value of x is 5. Question 2a) i): Given the matrix M = 3 & 2 \\ 5 & x . Find the value of x for which the determinant is 23. Step 1: Calculate the determinant of M. (M) = (3)(x) - (2)(5) = 3x - 10 Step 2: Set the determinant equal to 23 and solve for x. 3x - 10 = 23 3x = 23 + 10 3x = 33 x = (33)/(3) x = 11 The value of x is 11. Question 2a) ii): Hence find the inverse of M. Step 1: Substitute the value of x found in part 2a) i) into matrix M. From 2a) i), x=11. M = 3 & 2 \\ 5 & 11 The determinant of M is given as 23. Step 2: Calculate the inverse of M. For a 2 × 2 matrix a & b \\ c & d , the inverse is (1)/(ad-bc) d & -b \\ -c & a . M^-1 = (1)/(23) 11 & -2 \\ -5 & 3 The inverse of M is (1)/(23) 11 & -2 \\ -5 & 3 . Question 2b) i): Given that Matrix A = 4 & -5 \\ 1 & 2 and B = 7 & y \\ 3 & 5 . Find the value of y for which the determinant of A and B are equal. Step 1: Calculate the determinant of matrix A. (A) = (4)(2) - (-5)(1) = 8 - (-5) = 8 + 5 = 13 Step 2: Calculate the determinant of matrix B. (B) = (7)(5) - (y)(3) = 35 - 3y Step 3: Set the determinants equal and solve for y. (A) = (B) 13 = 35 - 3y 3y = 35 - 13 3y = 22 y = (22)/(3) The value of y is (22)/(3). Question 2b) ii): Hence find the inverse of B. Step 1: Substitute the value of y found in part 2b) i) into matrix B. From 2b) i), y=(22)/(3). B = 7 & (22)/(3) \\ 3 & 5 The determinant of B is 13 (since (A) = (B) and (A)=13). Step 2: Calculate the inverse of B. B^-1 = (1)/(13) 5 & -(22)/(3) \\ -3 & 7 You can also distribute the (1)/(13): B^-1 = (5)/(13) & -(22)/(39) \\ -(3)/(13) & (7)/(13) The inverse of B is (1)/(13) 5 & -(22)/(3) \\ -3 & 7 . Drop the next question!