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A square matrix is invertible exactly when its determinant is nonzero. If the determinant is zero, the matrix is singular and not invertible.

Conditions: The matrix is square.; The determinant has been calculated.

The word ALOMOMOLA has 9 letters. It contains two 'A's, two 'L's, two 'M's, and three 'O's.

Conditions: Word length is 9.; Letter frequencies: $A=2$, $L=2$, $M=2$, $O=3$.

The joint probability P(A and B) is equal to the product of the individual probabilities $P(A)P(B)$ if and only if the events A and B are independent. Independence means that the occurrence of one event does not affect the probability of the other, which is mathematically expressed as P(B|A) = $P(B)$ (assuming $P(A) > 0$).

Conditions: Events A and B are independent.; For the conditional equality P(B|A) = $P(B)$, event A must have positive probability.

The expected number of heads is 50 because the expected value for a counting experiment like this is calculated as the number of trials multiplied by the probability of success on each trial. With 100 tosses and a fair coin (probability of heads = 0.5), the calculation is $100 \cdot 0.5 = 50$.

Conditions: The coin is fair.; There are 100 tosses per trial.; $X$ counts heads.

No. Conditional independence does not imply marginal independence.

Conditions: The model has a common parent structure.; The child nodes are conditionally independent given the parent.

The time complexity of Dijkstra's algorithm is $O(|E| + |V|\log|V|)$ when implemented using a Fibonacci heap for the priority queue. Here, $|V|$ is the number of vertices and $|E|$ is the number of edges.

Conditions: The implementation uses a Fibonacci heap data structure for the priority queue.; $|V|$ represents the number of vertices.; $|E|$ represents the number of edges.

The derivative of $1 + e^u$ is $e^u$, which appears as a factor in the integrand $e^u \sqrt{1+e^u}$. This allows the entire term $e^u du$ to be replaced by $dx$, simplifying the integral to $\int \sqrt{x} dx$.

Conditions: The integrand is $e^u \sqrt{1+e^u}$.; The substitution is $x = 1 + e^u$.; The derivative $dx/du = e^u$ is present in the integrand.

A 2-Opt swap improves a route by selecting two edges in the current route, removing them, and reconnecting the four endpoints in a different way to form a new, valid cycle. This operation is particularly effective at eliminating self-intersections in a route, thereby reducing the total distance.

Conditions: Requires an existing valid TSP route.; Involves selecting two distinct edges.; The reconnection must produce a single valid cycle.

The standard rule (smaller denominator implies larger fraction) applies strictly when the common numerator is positive. If the common numerator is zero, all such fractions are equal to zero, resulting in equality.

Conditions: Comparing fractions $\frac{n}{a}$ and $\frac{n}{b}$ with $0 < a < b$; Case 1: $n > 0$ (standard case); Case 2: $n = 0$; Case 3: $n < 0$

Acceleration is defined as the second derivative of the position (or displacement) function $s(t)$ with respect to time $t$. It represents the rate of change of velocity, where velocity itself is the first derivative of displacement.

Conditions: Motion occurs along a fixed line with a chosen coordinate direction.; The position function $s(t)$ is twice differentiable with respect to time.

The definition requires $x \neq a$ (expressed as $0 < |x - a|$) because the limit describes the behavior of the function *as it approaches* the point $a$, not its value *at* the point $a$. The function might be undefined at $x = a$, or its value $f(a)$ might differ from the limit $L$.

Conditions: The limit is concerned with the trend of $f(x)$ near $a$.; $f(a)$ may be undefined or discontinuous at $a$.

Giving a column a nonzero component in a previously unused dimension makes the columns linearly independent, thereby increasing the exact rank to full rank. Even if this perturbation is tiny, the exact mathematical rank changes, although the columns remain nearly dependent.

Conditions: The original columns are linearly dependent.; A perturbation adds a nonzero component to a column in a new dimension.