Gauth vs Wolfram Alpha

Wolfram takes the app, the data, the plots, the practice generator, and — the row worth dwelling on — the engine itself. A computed answer and a written answer are not the same kind of object. Ours should be checked; theirs is the check.

Wolfram Alpha. Basic is free and does not include step-by-step solutions; Wolfram says so on the pricing page rather than letting you discover it mid-problem. As of September 2026 the student pricing page showed Pro at $9.99 a month struck through and $5.00 a month for verified students, with Pro Premium at $12.00 discounted to $8.25. Annual billing and student verification both change what you pay, so the plan you are shown depends on who you are. Pro buys step-by-step solutions, the Problem Generator, guided calculators, extended computation time, file uploads with instant analysis, and photo input in the iOS and Android apps.

Gauth. The first solution is free with no account and no card, and it includes the working. Free accounts get three solutions a day. Gauth Plus is $11.99 a month with a three-day free trial, $31.99 a quarter, or $99.99 a year, as of September 2026. Follow-up questions are included and do not consume quota. The pricing page lists the current plans.

For a verified student, Wolfram Alpha Pro at $5.00 a month is less than half the price of Gauth Plus and comes with a much larger machine behind it. The reason to pay us instead is the explanation, and if you do not need the explanation you should not pay for it.

Find the eigenvalues and eigenvectors of $A=\left(\begin{array}{cc}2& 1\\ 1& 2\end{array}\right)$. This is a standard linear algebra exercise where the answer is short and the procedure is the entire point.

Wolfram Alpha answers this instantly and for free, and gives you far more than you asked for: eigenvalues 3 and 1, the corresponding eigenvectors, the characteristic polynomial, the determinant and trace, a diagonalisation, and often a geometric picture of what the matrix does to the plane. Everything is exact rather than numerical. What the free tier will not show you is the derivation — the step-by-step panel is behind Pro — so if you cannot already produce $\det (A-\lambda I)=0$ yourself, the free result is a destination with no route to it. That is a deliberate product decision, clearly labelled, not a trick.

Gauth spends its answer on the route. It sets up $\det (A-\lambda I)=0$ and says why that equation is the definition of an eigenvalue rather than a formula to memorise, expands to ${\lambda }^2-4\lambda +3$, factors, and then solves the two homogeneous systems, noting the thing that confuses nearly everyone the first time: the system is deliberately singular, so an eigenvector is only ever determined up to a scale factor, and $(2,2)$ is as correct as $(1,1)$. You get no plot and no diagonalisation unless you ask, but you can ask. If the surrounding machinery is what is missing, inverting a matrix two ways covers the row operations this leans on.

One practical note if you stay on the free tier: that dense results page is itself a checking instrument. Wolfram prints the trace and the determinant alongside the eigenvalues, and for any 2×2 matrix the eigenvalues must sum to the trace and multiply to the determinant. Here $3+1=4$ and $3\times 1=3$, which matches. You can verify a derivation you did by hand against the free output in about five seconds without ever unlocking a single step.

Wolfram Alpha is not a homework product with a solver attached; it is the Wolfram Language and a curated knowledge base with a search box in front of them. That architecture buys exactness. Ask for a closed form and you get a closed form, not a decimal approximation. Ask for a value to fifty digits and you get fifty digits. Ask something ill-posed and it tells you what it assumed. For anyone whose work has to be right rather than merely convincing, that is the correct tool, and it is the reason researchers keep it open in a tab long after their last exam.

The curated data is the second advantage and it has no equivalent here. Physical constants with their uncertainties, unit conversions that carry dimensions through a calculation, chemical properties, astronomical positions, economic series, population figures. You can type a physics question that mixes a measured quantity with a constant and get a dimensionally consistent answer, which is exactly the kind of problem where a written solution can quietly go wrong.

Even the free tier gives you things Gauth does not have at any price: a plot of the function next to the answer, alternate forms of an expression, roots and asymptotes listed without being asked, series expansions, and a domain and range summary. Seeing the alternate form of an expression is often what makes a problem click, and Wolfram shows it by default rather than on request.

Pro adds the Problem Generator, which produces practice questions with hints and full solutions in the topic you were just working on, plus guided calculators that walk you through a specific procedure. And for a verified student, Pro at $5.00 a month as of September 2026 undercuts Gauth Plus substantially. If you can already read a derivation and what you need is a trustworthy engine, buy the engine. For the moments when the derivation itself is the problem, the linear algebra solver here writes out every row operation instead.