Does the description disclose side effects, auth requirements, rate limits, or destructive behavior?

Annotations already cover the safety profile (readOnly, idempotent, non-destructive, closed-world), so the description's added value is the exactness guarantee ('fraction-preserving elimination') and the disclosure that output includes both the determinant and detailed operation steps. It does not mention behavior on non-square or malformed input, which is the remaining gap.

Agents need to know what a tool does to the world before calling it. Descriptions should go beyond structured annotations to explain consequences.

Is the description appropriately sized, front-loaded, and free of redundancy?

Three sentences, front-loaded with the core action, followed by use cases and return content. The middle sentence is a somewhat padded feature list, but nothing is seriously wasteful.

Shorter descriptions cost fewer tokens and are easier for agents to parse. Every sentence should earn its place.

Given the tool's complexity, does the description cover enough for an agent to succeed on first attempt?

With no output schema, the description correctly compensates by describing the return value (determinant plus operation steps). For a single-parameter, annotated, read-only computation the definition is nearly complete; only error/edge-case behavior is unstated.

Complex tools with many parameters or behaviors need more documentation. Simple tools need less. This dimension scales expectations accordingly.

Does the description clarify parameter syntax, constraints, interactions, or defaults beyond what the schema provides?

Only one parameter, and schema description coverage is 100% with the schema stating 'Square matrix.' The description repeats the same constraint ('square matrix') without adding format or element-type detail (e.g. strings accepted for exact fractions). With the schema doing the heavy lifting, baseline 3 applies.

Input schemas describe structure but not intent. Descriptions should explain non-obvious parameter relationships and valid value ranges.

Does the description clearly state what the tool does and how it differs from similar tools?

States a specific verb ('Find the exact determinant') and resource ('square matrix'), plus the technique used ('fraction-preserving elimination'). No sibling (add, inverse, multiply, subtract, transpose) computes a determinant, so the agent can route unambiguously without opening any schema.

Agents choose between tools based on descriptions. A clear purpose with a specific verb and resource helps agents select the right tool.

Does the description explain when to use this tool, when not to, or what alternatives exist?

The 'Use for determinant questions requiring row swaps, elimination steps, pivot multipliers, sign changes, diagonal products, or singularity checks' clause gives concrete invocation context. However, it enumerates sub-features rather than naming alternatives or stating exclusions; the routing benefit comes mostly from the non-overlapping sibling set rather than from explicit guidance.

Agents often have multiple tools that could apply. Explicit usage guidance like "use X instead of Y when Z" prevents misuse.