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If the matrix $P = \begin{pmatrix}1 & \alpha & 3\\1 & 3 & 3\\2 & 4 & 4\end{pmatrix}$ is the adjoint of a $3\times3$ matrix $A$ with $|A|=4$, find $\alpha$.
We use the property: $|adj(A)| = |A|^{n-1}$ for an $n\times n$ matrix. Here $n=3$, so $|P| = |A|^2 = 16$.
Compute $|P|$ by expanding along row 1 (with unknown $\alpha$):
$|P| = 1(3\cdot4-3\cdot4) - \alpha(1\cdot4-3\cdot2) + 3(1\cdot4-3\cdot2)$
$= 1(0) - \alpha(4-6) + 3(4-6) = 2\alpha - 6$
Setting $2\alpha - 6 = 16 \Rightarrow 2\alpha = 22 \Rightarrow \alpha = \mathbf{11}$
Comparing the business's records with the supplier's statement is the most effective way to find missing credit notes, as the supplier will have recorded the credit note even if the business forgot to. A trial balance or control account only checks internal consistency.
The Cash Conversion Cycle formula is:
$\text{CCC} = \text{Days Inventory Outstanding} + \text{Days Sales Outstanding} - \text{Days Payable Outstanding}$
$= 60 + 30 - 40 = \mathbf{50 \text{ days}}$
The CCC measures how long it takes a firm to convert its investments in inventory and other resources into cash flows from sales. A shorter CCC generally indicates better liquidity management.
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