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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.
Acute Epiglottitis β a life-threatening supraglottic infection (typically Haemophilus influenzae type b in unvaccinated children).
Classic presentation:
- Sudden high fever, toxic appearance
- Drooling (inability to swallow secretions)
- Muffled 'hot potato' voice
- Tripod/sniffing position: leaning forward, neck extended, supporting on hands β maximizes airway opening
- Stridor (late, ominous sign)
- Anxiety, refuses to lie down
CRITICAL NURSING RULE:
NEVER examine the throat with tongue depressor, attempt IV access, take blood, agitate the child, or lay the child supine β ANY stimulation can cause complete airway obstruction.
Priority actions:
- Keep child calm, upright (let child maintain own position)
- Administer humidified Oβ gently
- Notify physician and anesthesiologist IMMEDIATELY
- Prepare for emergency intubation or tracheostomy (bedside)
- IV access and antibiotics ONLY after airway is secured
Treatment: Ceftriaxone \(100\,\text{mg/kg/day}\) IV after airway secured.
Differentiate from croup: croup has barking cough, subglottic involvement, not toxic, managed with steroids + nebulized epinephrine.
Down the group, bond length increases (HβX bond gets weaker), making it easier to release HβΊ. Therefore acid strength increases down the group:
\(\text{H}_2\text{O} < \text{H}_2\text{S} < \text{H}_2\text{Se} < \text{H}_2\text{Te}\)
Note: This trend is opposite to electronegativity trend, and is governed by bond dissociation energy.
Work done at constant pressure is given by $W = P \Delta V$.
Using the ideal gas law ($PV = nRT$), at constant pressure, $P \Delta V = n R \Delta T$.
$W = 0.5 \times 8.31 \times (90 - 20)$
$W = 0.5 \times 8.31 \times 70 = 35 \times 8.31 = 290.85 \text{ J}$.
Rounding gives approximately $291 \text{ J}$.
Birth-related Brachial Plexus Injuries:
| Type | Nerve Roots | Presentation | Cause |
|---|---|---|---|
| Erb's Palsy | C5, C6 | 'Waiter's tip': arm adducted, internally rotated, elbow extended, wrist pronated, fingers flexed | Shoulder dystocia (excessive lateral neck traction) |
| Klumpke's Palsy | C8, T1 | Claw hand deformity (intrinsic hand muscles paralyzed); Horner's syndrome if T1 root avulsed | Arm hyperabduction during delivery |
| Total Plexus Palsy | C5βT1 | Flail arm, complete paralysis | Severe traction |
'Waiter's tip' = Erb's Palsy (C5βC6). Absent Moro on the affected side (normal Moro requires shoulder abduction/external rotation via C5βC6).
Nursing management:
- Gentle range of motion exercises after 7β10 days
- Proper positioning (avoid abduction)
- Physiotherapy referral
- Most recover spontaneously in 3β6 months
- EMG/nerve conduction at 3 months if no improvement
- Surgical repair (nerve grafting) if no recovery by 3β9 months
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