Matrix Calculator

The Matrix Calculator is a free online tool that perform matrix addition, multiplication, determinant, inverse, transpose, rref, rank, and power up to 6x6 with optional step-by-step solutions.. It runs entirely in your browser with no installation or sign-up needed.

Addition, subtraction, multiplication, scalar multiplication, power, transpose, determinant, inverse, trace, rank, and RREF for matrices up to 6x6.

Yes — completely free, no registration. All math runs client-side in your browser.

Yes. Enable the step-by-step toggle to see cofactor expansion for determinants and full Gauss-Jordan row operations for matrix inverses.

Absolutely. The Matrix Calculator processes everything client-side in your browser. No data is uploaded to or stored on any server. Your content remains private on your device at all times.

Yes, the Matrix Calculator is fully responsive and works on smartphones and tablets. You can use it on any device with a modern web browser -- no app download required.

No account or registration is needed. Simply open the Matrix Calculator in your browser and start using it immediately. There are no sign-up walls or usage restrictions.

The Matrix Calculator uses industry-standard formulas and algorithms to ensure accurate results. However, for critical financial or medical decisions, always consult a qualified professional.

Simply enter your input in the provided field, adjust any settings to your preference, and the tool will process it instantly. You can then copy the result to your clipboard or download it.

The Matrix Calculator works in all modern browsers including Chrome, Firefox, Safari, Edge, and Opera. For the best experience, use the latest version of your preferred browser.

Matrix multiplication only works when the number of columns in the first matrix equals the number of rows in the second. So a 2x3 matrix can multiply a 3x4 matrix, because the inner dimensions (3 and 3) match, and the result is 2x4 — taking the outer dimensions. If those inner numbers disagree, the dot products between rows and columns aren't defined, and the operation is impossible. This is different from addition and subtraction, which instead require both matrices to be exactly the same shape. Multiplication is also not commutative: A×B and B×A usually give different answers, and B×A may not even be valid. When the dimensions don't line up, this calculator tells you exactly why rather than guessing. Enter your matrices, pick A×B, and it checks the shapes and computes the product up to 6x6 instantly.

A square matrix is singular when its determinant equals zero, and a singular matrix has no inverse — there is no matrix you can multiply it by to get the identity. Geometrically, a singular matrix collapses space onto a lower dimension, so the transformation can't be undone; information is lost and can't be recovered. Only square matrices can have an inverse at all, and among those, exactly the ones with a non-zero determinant are invertible. A zero determinant also signals that the matrix's rows or columns are linearly dependent, meaning its rank is less than its size. This calculator computes the inverse by Gauss-Jordan elimination and, if it hits a zero pivot, stops and reports that the matrix is singular instead of returning garbage. Enter a square matrix and choose inverse to check invertibility directly.

The rank of a matrix is the number of linearly independent rows or columns it has, and reducing the matrix to reduced row echelon form (RREF) is the standard way to find it. In RREF, each leading entry is a 1 with zeros above and below it, and the rank is simply the count of non-zero rows that remain — equivalently, the number of pivot positions. Rows that reduce entirely to zero contribute nothing, signalling dependence among the original rows. Rank tells you how many independent equations a system really has, whether a set of vectors spans the space you expect, and whether a square matrix is full-rank and therefore invertible. This calculator does both: pick RREF(A) to see the reduced matrix, or rank(A) to get the number directly, for any matrix up to 6x6.

Many matrix operations, especially inverses and RREF, produce values that aren't whole numbers, and rounding them to decimals introduces small errors that compound through later steps. A decimal like 0.3333 is really one-third truncated, so writing it as 1/3 keeps the value exact. This calculator detects clean rational results with small denominators (up to 16) and displays them as proper fractions instead of long decimals, which matches how textbooks and homework expect answers to look and makes it easy to compare against a hand calculation. Exact fractions are particularly valuable when you invert a matrix and want to verify that A times its inverse really returns the identity without floating-point drift. If a value isn't a clean fraction, it falls back to a decimal. Run an inverse or RREF here and read the exact fractional answer.

The trace of a square matrix is the sum of the entries on its main diagonal — top-left to bottom-right. For example, a matrix with diagonal entries 4, 1, and 7 has a trace of 12. Only square matrices have a trace, since non-square matrices don't have a complete main diagonal. Despite being a simple sum, the trace carries real meaning: it equals the sum of the matrix's eigenvalues, it stays the same under similarity transformations, and it appears throughout statistics, physics, and machine learning, for instance as the total variance in a covariance matrix. It's also a quick sanity check, since trace is unchanged by transposing a matrix. This calculator computes the trace of any square matrix up to 6x6 in one click — enter your matrix, select tr(A), and read the diagonal sum instantly.