Build a rectangular matrix before choosing the calculation
A matrix is organized by position: an m × n matrix has m horizontal rows, n vertical columns, and an entry aᵢⱼ at row i and column j. On this route, select the row and column counts for A and, when needed, B before calculating. The text editor must contain exactly that rectangle. Put each row on a new line or separate rows with semicolons; separate entries with spaces or commas. Blank, ragged, nonnumeric, nonfinite, or oversized input is rejected instead of being padded or guessed.
That dimension declaration is part of the mathematics, not a display preference. The same sequence of six values could describe a 2 × 3 matrix or a 3 × 2 matrix, and those objects support different products and transformations. The editor resizes from 1 × 1 through 10 × 10 and shows the selected shape beside each matrix. If the task is specifically to inspect every dot product in AB with linked A-columns and B-rows, the matrix multiplication calculator provides a focused multiplication ledger; this route remains useful when the next question may instead be addition, transpose, determinant, inverse, RREF, trace, or A².
Let dimensions decide whether addition, subtraction, or multiplication exists
The operation selector checks compatibility before it evaluates entries. Similar-looking arrays are not automatically compatible matrices.
Addition and subtraction compare matching positions
A + B and A − B require identical dimensions. For two 2 × 3 matrices, each result entry uses the two values in the same row and column, so the answer is also 2 × 3. A 3 × 2 matrix contains the same number of entries but cannot be added to a 2 × 3 matrix because positions do not align.
Multiplication matches the inner dimensions
If A is m × n and B is n × p, then AB exists and has shape m × p. Entry (i,j) is the dot product of row i of A with column j of B. The page reports the readiness rule before calculating and shows concrete dot-product work in the result, rather than multiplying entries position by position.
Order is part of a matrix product
AB and BA ask different questions. One may exist when the other does not, and when both exist their shapes or entries can differ. Squaring A means the ordered product A × A and therefore requires a square matrix; it is not the operation of squaring every individual entry.
Unary operations have their own shape rules
A scalar k multiplies every entry and accepts any supported rectangle. Transpose swaps rows with columns, turning m × n into n × m. Trace adds only the main diagonal and therefore requires a square matrix. These results answer different structural questions and should not be substituted for one another merely because they return a matrix or a number.
Separate determinant evidence from the inverse calculation
The determinant is a signed scalar defined here only for a square matrix. A nonzero determinant means the square matrix is nonsingular in exact linear algebra; a zero determinant means its rows or columns are linearly dependent and it has no two-sided inverse. The determinant is not the sum of all entries, and its sign and magnitude do not by themselves solve a system or describe every geometric property of the transformation.
The inverse A⁻¹ is another square matrix satisfying AA⁻¹ = A⁻¹A = I. This calculator obtains it by row-reducing the augmented block [A | I]. If it cannot find a full pivot set, it stops with a singular-matrix error instead of displaying divided-by-zero entries. When the result contains decimals that correspond to simple rational values, the fraction calculator can reduce a separately copied scalar ratio, but it cannot decide whether the original matrix is invertible.
Determinant zero and inverse failure are therefore related checks, not two interchangeable output formats. For a nearly singular matrix, small changes in entered decimal values can produce large changes in the inverse even when the computed determinant is not exactly zero. Preserve the original entries and inspect the reported elimination steps before interpreting a large inverse coefficient as stable information.
Encode a linear system as an augmented matrix and interpret every pivot
RREF accepts rectangular input, so it can expose the structure of a system when the last column is deliberately entered as the right-hand side.
- Keep variable columns and constants in a declared order
For equations in x, y, and z, enter the x coefficients in the first column, y coefficients in the second, z coefficients in the third, and constants in the final column. A missing variable needs a zero coefficient. The editor does not read equation text or infer variable names, so record the column order beside the matrix.
- Use row operations that preserve the solution set
RREF swaps rows, scales a row by a nonzero value, and adds a multiple of one row to another. The result exposes pivot columns, free-variable columns, and contradictory rows. For a guided equation-first workflow that labels variables and classifies a supported system, use the system of equations calculator; do not assume that any final row of zeros means there is no solution.
- Distinguish unique, infinite, and inconsistent outcomes
A pivot in every variable column with no contradictory row gives a unique solution. A nonpivot variable column can create infinitely many solutions when the remaining rows are consistent. A row whose coefficient side is all zero but whose constant is nonzero represents a contradiction and therefore no solution.
- Do not force a one-variable problem into matrix notation
When the original task is one supported scalar equation rather than a simultaneous linear system, the solve equation calculator keeps the algebraic form and root checks visible. Matrix RREF is valuable only after the system has been translated into coefficients without losing signs, constants, or variable order.
Audit numerical matrix output within the page's stated boundary
A visible calculation trail helps reproduce the arithmetic, but it does not turn finite-precision output into a symbolic proof.
- Confirm the selected dimensions against the pasted rectangle before reading a result. A correct operation applied to a transposed or misordered data table still answers the wrong question.
- Read the first compatibility statement and a sample of the displayed entry formulas or elimination steps. For large matrices, detailed steps are intentionally bounded, so the page summarizes omitted repetitive work rather than emitting an unreviewable transcript.
- Treat displayed decimals as numerical approximations. Floating-point elimination uses scale-aware pivot checks, yet badly conditioned data can remain sensitive to measurement error and rounding. Independent residual checks such as AX − B or AA⁻¹ − I are appropriate for consequential work.
- The route handles finite real entries only. It does not provide exact symbolic fractions, complex matrices, eigenvalues, eigenvectors, decompositions, units, uncertainty propagation, saved cloud documents, collaboration features, or an external account interface.
- Choose the operation that matches the mathematical claim: transpose for orientation, determinant for one square-matrix scalar, inverse for a nonsingular square transformation, and RREF for pivot structure. A result from one mode is not evidence that every other operation is defined.