Given A = p & 2 -3 & p+5 .

a) Step 1: For a matrix A = a & b \\ c & d to be singular, its determinant must be zero. The determinant is ad - bc. Given A = p & 2 \\ -3 & p+5 . Set the determinant to zero: p(p+5) - (2)(-3) = 0 Step 2: Solve the equation for p. p^2 + 5p + 6 = 0 Factor the quadratic equation: (p+2)(p+3) = 0 This gives two possible values for p: p+2 = 0 p = -2 p+3 = 0 p = -3 The values of p are -2 or -3. b) i) Step 1: Let x be the price of a shirt and y be the price of a pair of trousers. From the first statement, 8 shirts and 5 pairs of trousers are valued at Ksh. 49,600. 8x + 5y = 49600 From the second statement, the number of shirts is reduced by 2 (to 8-2=6 shirts) and the number of pairs of trousers is increased by 2 (to 5+2=7 pairs of trousers), with a new value of Ksh. 52,800. 6x + 7y = 52800 Step 2: Form the matrix equation. 8 & 5 \\ 6 & 7 x \\ y = 49600 \\ 52800 ii) Step 1: To determine the price of each item, we need to solve the matrix equation from part (b)(i). Let M = 8 & 5 \\ 6 & 7 , X = x \\ y , and C = 49600 \\ 52800 . We need to find X = M^-1C. First, calculate the determinant of M: (M) = (8)(7) - (5)(6) = 56 - 30 = 26 Step 2: Calculate the inverse of M: M^-1 = (1)/(26) 7 & -5 \\ -6 & 8 Step 3: Multiply M^-1 by C to find X: x \\ y = (1)/(26) 7 & -5 \\ -6 & 8 49600 \\ 52800 x \\ y = (1)/(26) 7(49600) + (-5)(52800) \\ -6(49600) + 8(52800) x \\ y = (1)/(26) 347200 - 264000 \\ -297600 + 422400 x \\ y = (1)/(26) 83200 \\ 124800 Step 4: Calculate the values of x and y: x = (83200)/(26) = 3200 y = (124800)/(26) = 4800 The price of a shirt is Ksh. 3200 and the price of a pair of trousers is Ksh. 4800. Shirt: Ksh. 3200, Trousers: Ksh. 4800 iii) Step 1: Calculate the profit from selling one shirt and one pair of trousers. Cost of one shirt (x) = Ksh. 3200. Profit on shirt = 20%. Profit from shirt = 0.20 × 3200 = Ksh. 640. Cost of one pair of trousers (y) = Ksh. 4800. Profit on trousers = 25%. Profit from trousers = 0.25 × 4800 = Ksh. 1200. Step 2: Calculate the total cost and total profit for one shirt and one pair of trousers. Total cost = Cost of shirt + Cost of trousers = 3200 + 4800 = Ksh. 8000. Total profit = Profit from shirt + Profit from trousers = 640 + 1200 = Ksh. 1840. Step 3: Calculate the percentage profit. Percentage Profit = Total ProfitTotal Cost × 100\% Percentage Profit = (1840)/(8000) × 100\% Percentage Profit = 0.23 × 100\% = 23\% The percentage profit made from selling a shirt and a pair of trousers is 23\%. Send me the next one 📸