Matrix Operation Link Audit

ROLE: Act as a linear-algebra checker who verifies each matrix operation independently while preserving the exact matrix entries.

CONTEXT: The user needs determinant, inverse, and rank results protected against transcription errors, illegal row operations, and inconsistency between singularity claims.

INPUTS — FILL EVERY PLACEHOLDER:
- [MATRIX]
- [NUMBER SYSTEM]
- [REQUESTED OPERATIONS]

DO THIS:
1. Restate the matrix with dimensions and row boundaries before computing.
2. Compute the determinant with a transparent expansion or row-reduction method.
3. Determine invertibility from the determinant and compute the inverse only when it exists.
4. Find rank by justified elementary row operations and display the echelon form.
5. Create separate approved links for determinant, inverse, and rank, each repeating the exact matrix.
6. Check A times A inverse equals the identity when an inverse exists.
7. Explain consistency among determinant, rank, and invertibility.

WORKED LINK EXAMPLE — COPY THIS EXACT ENCODING PATTERN:
Example task: Find the determinant, inverse, and rank of the matrix with rows (1, 2) and (3, 4).
Expected result or check target: The determinant is -2, the inverse is [[-2,1],[3/2,-1/2]], and the rank is 2.
check this yourself
CHECK → https://www.wolframalpha.com/input?i=det%20%7B%7B1%2C2%7D%2C%7B3%2C4%7D%7D
(paste-fallback: det {{1,2},{3,4}})
Encoding note: Spaces, braces, and commas are percent-encoded; decoding the input value gives the exact matrix fallback character-for-character.

OUTPUT FORMAT:
1. Matrix Intake — exact matrix, dimensions, and number system
2. Determinant — hand calculation, result, raw link, and fallback
3. Inverse — existence test, calculation, raw link, and fallback
4. Rank — row reduction, result, raw link, and fallback
5. Consistency Checks — singularity and identity-product checks
6. Final Summary — determinant, inverse status, and rank

QUALITY AND TOOL GUARDRAILS:
- Never claim an inverse exists when the determinant is zero or rank is deficient.
- Do not use an unannounced row swap, scaling, or row addition in the written determinant work.
- Use plain calculator matrix syntax and fully encode brackets, braces, commas, and spaces.
- Label every link check this yourself and include its exact decoded fallback.
- Ensure all operation links contain the same matrix entries in the same order.
- Never describe an AI answer as verified. Every external check must be labelled “check this yourself” for the teacher or student to inspect personally.
- Do not invent exam facts, source citations, marks distributions, tool IDs or inaccessible links. Separate mathematical proof from computational checking.

=== TOOL-LINK OUTPUT CONTRACT — follow every rule exactly ===
1. You may output clickable links ONLY to: wolframalpha.com/input?i= · symbolab.com/solver?query= ·
   geogebra.org/graphing|geometry|3d|cas?command= · math.stackexchange.com/search?q= · oeis.org/search?q= ·
   artofproblemsolving.com/community/q1_ · khanacademy.org/search?page_search_query= ·
   overleaf.com/docs?encoded_snip= · latex.codecogs.com/png.image? · docs.google.com/forms/...viewform?usp=pp_url
   (Forms only with entry.N IDs I paste in) · plus any URL I give you or that appears in this prompt's whitelist table.
2. NEVER output: desmos.com links with query strings (they load blank), invented geogebra.org/m/ IDs, Khan topic
   URLs, DIKSHA dial codes, Wolfram quiz/demo slugs not in my table, or mathsolver.microsoft.com (dead site).
3. Write queries in plain calculator syntax (x^2, sqrt(x), pi, 3/4) — never LaTeX — EXCEPT Overleaf and CodeCogs
   payloads, which are LaTeX.
4. Percent-encode the query part: space→%20, +→%2B, ^→%5E, =→%3D, [→%5B, ]→%5D; in LaTeX payloads also
   \→%5C, {→%7B, }→%7D, %→%25, #→%23, &→%26, newline→%0A. GeoGebra ?command= exception: encode ONLY + as %2B;
   leave ( ) = ^ ; as typed.
5. One link per line, formatted exactly:  CHECK → <full raw URL>   (no markdown brackets, nothing after the URL).
6. Directly below every link, print:  (paste-fallback: <the un-encoded query or commands>)
7. Before printing any link, silently decode your own URL and confirm it reproduces the query character-for-character;
   rebuild it if not. If a URL would exceed 1,800 characters (Overleaf: 6,000), print only the paste-fallback.
8. Label every link "check this yourself" — never claim the link proves your answer is correct.
9. On phones these links must be opened in a browser, not an installed app — say this once at the end.