Given A = p & 2 -3 & p+5 .
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
a)
Step 1: For a matrix $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ to be singular, its determinant must be zero. The determinant is $ad - bc$.
Given $A = \begin{pmatrix} p & 2 \\ -3 & p+5 \end{pmatrix}$.
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 $p$:
$$p+2 = 0 \implies p = -2$$
$$p+3 = 0 \implies p = -3$$
The values of $p$ are $\boxed{-2 \text{ or } -3}$.
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.
$$\begin{pmatrix} 8 & 5 \\ 6 & 7 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 49600 \\ 52800 \end{pmatrix}$$
ii)
Step 1: To determine the price of each item, we need to solve the matrix equation from part (b)(i).
Let $M = \begin{pmatrix} 8 & 5 \\ 6 & 7 \end{pmatrix}$, $X = \begin{pmatrix} x \\ y \end{pmatrix}$, and $C = \begin{pmatrix} 49600 \\ 52800 \end{pmatrix}$.
We need to find $X = M^{-1}C$. First, calculate the determinant of $M$:
$$\det(M) = (8)(7) - (5)(6) = 56 - 30 = 26$$
Step 2: Calculate the inverse of $M$:
$$M^{-1} = \frac{1}{26} \begin{pmatrix} 7 & -5 \\ -6 & 8 \end{pmatrix}$$
Step 3: Multiply $M^{-1}$ by $C$ to find $X$:
$$\begin{pmatrix} x \\ y \end{pmatrix} = \frac{1}{26} \begin{pmatrix} 7 & -5 \\ -6 & 8 \end{pmatrix} \begin{pmatrix} 49600 \\ 52800 \end{pmatrix}$$
$$\begin{pmatrix} x \\ y \end{pmatrix} = \frac{1}{26} \begin{pmatrix} 7(49600) + (-5)(52800) \\ -6(49600) + 8(52800) \end{pmatrix}$$
$$\begin{pmatrix} x \\ y \end{pmatrix} = \frac{1}{26} \begin{pmatrix} 347200 - 264000 \\ -297600 + 422400 \end{pmatrix}$$
$$\begin{pmatrix} x \\ y \end{pmatrix} = \frac{1}{26} \begin{pmatrix} 83200 \\ 124800 \end{pmatrix}$$
Step 4: Calculate the values of $x$ and $y$:
$$x = \frac{83200}{26} = 3200$$
$$y = \frac{124800}{26} = 4800$$
The price of a shirt is Ksh. 3200 and the price of a pair of trousers is Ksh. 4800.
$$\boxed{\text{Shirt: Ksh. 3200, Trousers: Ksh. 4800}}$$
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 = $0.20 \times 3200 = \text{Ksh. } 640$.
Cost of one pair of trousers ($y$) = Ksh. 4800. Profit on trousers = 25%.
Profit from trousers = $0.25 \times 4800 = \text{Ksh. } 1200$.
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 = $3200 + 4800 = \text{Ksh. } 8000$.
Total profit = Profit from shirt + Profit from trousers = $640 + 1200 = \text{Ksh. } 1840$.
Step 3: Calculate the percentage profit.
$$\text{Percentage Profit} = \frac{\text{Total Profit}}{\text{Total Cost}} \times 100\%$$
$$\text{Percentage Profit} = \frac{1840}{8000} \times 100\%$$
$$\text{Percentage Profit} = 0.23 \times 100\% = 23\%$$
The percentage profit made from selling a shirt and a pair of trousers is $\boxed{23\%}$.
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