Matrix Calculator

Put each row on its own line and separate the entries with spaces or commas, for example 1 2 on the first line and 3 4 on the second. You can also separate rows with semicolons, as in 1 2; 3 4, or paste bracket notation such as [[1, 2], [3, 4]]. Every row needs the same number of entries.

Entries can be integers, decimals, numbers in scientific notation such as 2.5e-3, fractions such as 3/4 and repeating decimals such as 0.(3). Each entry is converted to an exact fraction before any calculation. Expressions such as sqrt(2) or pi are not accepted as entries, so use their decimal values instead.

Each matrix can have up to 10 rows and 10 columns, and each input can be up to 2000 characters long. Larger matrices are rejected with a size message. The limit keeps exact fraction arithmetic fast, because the numerators and denominators of an inverse can grow quickly.

The calculator works with exact rational numbers, so the inverse of [[1, 2], [3, 4]] is shown as -2, 1, 3/2 and -1/2 instead of a rounded decimal. Whenever the result contains a non-integer value, a second block shows the same matrix as decimals rounded to the decimal places you selected. Exact values avoid the rounding errors that floating point arithmetic adds.

Addition, subtraction and multiplication use Matrix B, and the calculator asks for it when it is empty. Determinant, inverse, transpose and trace work on Matrix A alone and ignore Matrix B. The operation menu shows which matrices each choice uses.

A + B and A - B require both matrices to have exactly the same number of rows and columns. A × B requires the number of columns in A to equal the number of rows in B, and the result has the rows of A and the columns of B. The tool explains which rule was broken when the sizes do not fit.

Only square matrices with a nonzero determinant have an inverse. When the determinant is exactly zero the matrix is singular, and the tool reports that instead of returning a matrix of huge numbers. Because the determinant is computed exactly, the check is not affected by rounding.

The trace is the sum of the entries on the main diagonal of a square matrix. For [[1, 2], [3, 4]] the trace is 1 + 4 = 5. It equals the sum of the eigenvalues and does not change when the matrix is transposed.

The tool uses the mathjs det function on exact fractions, which reduces the matrix step by step and multiplies the pivots. A 1 by 1 matrix returns its single entry and a 2 by 2 matrix returns ad - bc. Determinant, inverse and trace all require a square matrix.

No, A × B and B × A are usually different, and one of them may not even be defined for the given sizes. The tool always multiplies Matrix A on the left by Matrix B on the right. Swap the two inputs if you need B × A.