Educational tool; results are estimates for learning, verify with your textbook or instructor.
This toolkit works with two $2\times 2$ matrices $A$ and $B$. You can ask for a determinant, an inverse when it exists, or the products $AB$ and $BA$. Every entry is an ordinary number.
The numbers are meaningful when you are studying a linear map on the plane, a $2\times 2$ Jacobian, or a tiny system $A\mathbf{x}=\mathbf{b}$. In those settings the determinant tells you about oriented area scaling and invertibility, and the products show that order matters.
This is not a computer algebra system for $n\times n$ matrices. It does not compute eigenvalues on this page, it does not row reduce a general system, and it does not prove a theorem. For stretch factors, use the eigenvalue tool. For the idea of a derivative matrix, read the Jacobian explainer.
How to use the toolkit
Fill the four entries of $A$ and, if you need products or a second determinant, the four entries of $B$. The default $A$ is $\begin{pmatrix}1&2\\3&4\end{pmatrix}$, whose determinant is $-2$. Choose an operation from the menu and press Compute.
Determinant of $A$ is the fastest self check. Inverse of $A$ uses the adjugate formula $(1/\det A)\begin{pmatrix}d&-b\\-c&a\end{pmatrix}$. Product $AB$ multiplies in that order. Product $BA$ reverses the order on purpose so you can see they need not match.
How to read the output
A determinant is a single number. If it is zero, $A$ is singular: columns are linearly dependent, the map collapses area, and no inverse exists. If it is nonzero, the inverse option should return a matrix that multiplies with $A$ to something near the identity.
Products print as a $2\times 2$ array. Compare $AB$ and $BA$ on the default matrices. They disagree, which is the whole lesson. Matrix multiplication is composition of maps, and composition is not commutative in general.
Common pitfalls
A determinant near zero is a warning, not a rounding trophy. The inverse formula divides by $\det A$. If that number is tiny, entries of $A^{-1}$ can explode, and floating point will look confident while being fragile.
Do not read $AB$ as “$A$ and then $B$” without fixing a convention. Here $AB$ means $A$ acts after $B$ on a column vector, the usual textbook order. If your course writes maps on the left, this is the matching product.
The page only knows $2\times 2$ arrays. A Jacobian of a map $\mathbb{R}^2\to\mathbb{R}^2$ fits. A $3\times 3$ rotation does not. Do not paste a larger matrix into four boxes and hope.
A 3 by 3 determinant and inverse, with each minor written out in exact fractions, is on the 3 by 3 matrix calculator.
To row reduce a system and read whether it has one solution, infinitely many, or none, use the reduced row echelon form calculator.
Related reading
The idea of a derivative matrix is in Jacobian matrices in plain language and the Jacobian glossary entry. Symbolic partials of a formula are the Jacobian matrix calculator. Orthogonal columns are the Gram Schmidt calculator, and elimination factors are the LU decomposition calculator. Determinants and linear maps live on the linear algebra pillar and in the determinant, matrix, and linear map entries.
A study order is the linear algebra first course pathway. Next tool: 2 by 2 eigenvalue calculator. For a 3 by 3 inverse with steps, use the 3 by 3 matrix calculator. Stretch factors of that larger matrix, including a real eigenvector, are on the 3x3 eigenvalue calculator. Nearby geometry: vector operations calculator.
