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Let $A$ and $B$ be $3\times3$ real matrices where $A$ is symmetric and $B$ is skew-symmetric. The system $(A^2B^2 - B^2A^2)X = O$ (where $X$ is $3\times1$ and $O$ is the $3\times1$ zero matrix) has:
Let $C = A^2B^2 - B^2A^2$. Since $A$ is symmetric: $A^T=A$, so $(A^2)^T=A^2$. Since $B$ is skew-symmetric: $B^T=-B$, so $(B^2)^T=B^2$.
$C^T = (A^2B^2)^T - (B^2A^2)^T = B^2A^2 - A^2B^2 = -C$
So $C$ is skew-symmetric. A real skew-symmetric $3\times3$ matrix always has determinant 0 (since $\det(C)=\det(C^T)=\det(-C)=-\det(C)$, giving $\det(C)=0$).
Since $\det(C)=0$, the homogeneous system $CX=O$ has infinitely many solutions.
Naegele's Rule formula:
\[\text{EDD} = \text{LMP} - 3\,\text{months} + 7\,\text{days} + 1\,\text{year}\]
Or equivalently: Add 9 months and 7 days to LMP
Calculation:
- LMP: 10th September
- Subtract 3 months: 10th June
- Add 7 days: 17th June
- Add 1 year (next year)
Therefore, EDD = 17th June (next year)
Normal pregnancy duration: \(280\,\text{days}\) or \(40\,\text{weeks}\) from LMP \(= 266\,\text{days}\) from conception.
Important caveat: Naegele's rule assumes a regular 28-day cycle with ovulation on Day 14. Adjust for irregular cycles accordingly. Ultrasound (especially first trimester CRL) is more accurate for dating.
Fraction of boys \(= \dfrac{12}{30} = \dfrac{2}{5}\)
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