Matrix Calculator with Steps

Enter a matrix and choose what to calculate: the determinant, the inverse, the rank, a product, a system of linear equations or a matrix equation. The calculator shows every step, and under the answer you can choose another method of solution. The entries can be whole numbers, fractions such as 1/3, decimals and square roots such as sqrt(3).

Examples

Determinant of a 3×3 matrix Determinant of a 4×4 matrix by cofactor expansion Inverse of a 3×3 matrix Inverse of a matrix with square roots Rank of a matrix Eigenvalues and eigenvectors Product of two matrices System of 3 equations by Cramer's rule System with infinitely many solutions Matrix equation AX = B

How to enter a matrix

• Choose the number of rows and columns, then type the entries. The arrow keys move between the entries and Enter calculates. An empty entry counts as 0.
• Fractions are written with a slash: 3/4, -1/2. A mixed number is written with a space: 1 1/2. Decimals such as 0.25 are turned into exact fractions.
• Square roots are written with sqrt: sqrt(3), 2sqrt(5), sqrt(2)/2, (1+sqrt(5))/2. The answers stay exact, with the roots in them.
• An entry can also be an expression with + - * / ^ and brackets, such as (2/3)^2.
• A matrix copied from a spreadsheet or from a text, with each row on its own line, can be pasted into any entry. The form [[1,2],[3,4]] is read as well.

How to share a solution

Every calculation is written into the address of the page, after the # sign. Copy the address from the browser, or the link “Link to this solution” under the answer, and send it to someone else: the page opens with the same matrix, the same method and all the steps.

Methods of solution

When a problem can be solved in more than one way, the methods are listed above the steps. Click one to see the same problem solved by it.

• Determinant: the rule of Sarrus for a 3×3 matrix, cofactor expansion along any row or column, or row reduction to a triangular matrix.
• Inverse matrix: Gauss–Jordan elimination of the matrix written next to the identity matrix, or the adjugate matrix made of the cofactors.
• Rank: row reduction to row echelon form, or minors that grow by one row and one column at a time.
• Eigenvalues and eigenvectors: the roots of the characteristic polynomial det(A − λI) = 0, exact with fractions and square roots, and the eigenvectors from the reduced row echelon form of A − λI, for matrices up to 4×4.
• System of linear equations: Gaussian elimination, Gauss–Jordan elimination, Cramer's rule or the inverse matrix. The number of equations and the number of unknowns can differ; the calculator tells whether the system has one solution, no solution or infinitely many, and then writes them with parameters.
• Matrix equations AX = B and XA = B: with the inverse matrix or by Gauss–Jordan elimination.
• The calculator also adds, subtracts and multiplies matrices, multiplies a matrix by a number, raises it to a power, transposes it and brings it to row echelon form or reduced row echelon form.

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