Write the condition next to the rule
In a short mathematical solution, the easiest line to omit is often the one that makes the next line valid. A cancelled factor must be nonzero. A theorem needs its domain assumptions. A convenient property of one class of matrices may fail outside that class.
Use this review method: underline the operation, list the condition it requires, and test whether the question supplies it. If the condition is missing, try a small counterexample before continuing. The aim is to make your reasoning inspectable, not to write a formal proof for every routine calculation.
A missing value does not settle a limit
For x different from 3, (x² − 9)/(x − 3) equals x + 3. Its limit as x approaches 3 is therefore 6. The original expression remains undefined at 3. Cancelling a factor simplified the nearby values; it did not silently fill the missing point.
If a new definition sets f(3) = 6, the extended function is continuous there. If it sets f(3) = 10, the limit is still 6 but continuity fails. Keep three questions separate: is the value defined; does the limit exist; do they agree? The limit and continuity sources below explain why these tests are distinct.
A diagonal shortcut needs the right matrix
For the upper-triangular matrix with first row (2, 5) and second row (0, −1), the characteristic determinant is (2 − λ)(−1 − λ). Its eigenvalues are 2 and −1. The zero below the diagonal is what removes the off-diagonal product.
Now change that zero to 1. The determinant becomes (2 − λ)(−1 − λ) − 5, so simply reading the diagonal no longer works. This is an efficient counterexample to the claim that every matrix has its diagonal entries as eigenvalues. The original triangular example also permits a quick trace and determinant check.
Build a notebook of boundaries, not just formulas
For each missed question, record the shortcut you attempted and the smallest change that breaks it. A useful note might say: “Diagonal entries give the eigenvalues for this triangular matrix; check the characteristic polynomial if the triangular structure disappears.” Include one successful example and one counterexample.
On the next attempt, state the condition before doing the calculation. If you can explain both why the original method works and why the altered example needs another method, you have a more informative observation than a repeated correct answer alone. These exercises test local reasoning; they do not measure your overall GRE Mathematics readiness.
Check the underlying concepts
The sources support these concepts. The numerical examples and suggested routines are original Keiko study guidance, with no promise of a particular score outcome.
- A function can have a limit at a point where its value is undefined. Calculus Volume 1, Section 2.3: The Limit Laws (checked 2026-09-13).
- Continuity at a point requires the function value to agree with its limit. Calculus Volume 1, Section 2.4: Continuity (checked 2026-09-13).
- An eigenvalue makes the characteristic determinant zero. MIT OCW 18.06SC Linear Algebra, session: Eigenvalues and Eigenvectors (checked 2026-09-13).
Put this into practice
Sources
Every source below was accessed on the listed date. Pages — especially department admissions pages — change without notice; the links go to the live versions.
- Calculus Volume 1, Section 2.3: The Limit Laws — OpenStax. Accessed 2026-08-03.
- Calculus Volume 1, Section 2.4: Continuity — OpenStax. Accessed 2026-08-03.
- MIT OCW 18.06SC Linear Algebra, session: Eigenvalues and Eigenvectors — MIT OpenCourseWare. Accessed 2026-08-03.
Related guides
Sources and corrections
Sources last checked 2026-09-13Every source cited on this page was checked on the date shown, and we update the page when a source changes. If something looks wrong, tell us and we'll recheck it.