Matrix calculator with steps: multiply, invert, determinant and more

This matrix calculator adds, subtracts, multiplies, transposes and raises matrices to powers, and finds determinants, inverses, reduced row echelon form (RREF) and eigenvalues, for matrices up to 5 by 5. Answers are exact, with fractions instead of rounded decimals, and every step of the working is shown. Pick an operation, type matrix A (and B if it needs one) and read the answer.

It opens on the eigenvalues of a 2 by 2 matrix. Open the full matrix calculator over this page to try your own.

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Matrix Calculator

Multiply, invert and find determinants and eigenvalues

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What the matrix calculator does

How to use the matrix calculator

1. Choose an operation from the chips at the top of the Matrices section.
2. Set the size of A with the rows and columns buttons, from 1 to 5. For square-only operations there is a single size button.
3. Type each entry. Whole numbers, decimals and fractions all work: type a slash to make a fraction such as 2/3. New cells start at 0.
4. If the operation uses B, set its size and fill it in too. For Power, type n in Power, n.
5. Read the answer, then the working underneath. If something can't be done, such as multiplying a 2 by 3 by a 2 by 3, the calculator says why.
6. For a quick start, the Examples section loads a ready-made multiplication, inverse, 4 by 4 determinant, row reduction, eigenvalue problem, rotation or shear.

Entries must be numbers or fractions: no letters or square roots. Practice questions on determinants and products, with worked solutions, are under Random problem in the Practice section.

Worked examples you can check in the calculator

Example 1: matrix addition

Add matrices of the same size by adding the entries in matching positions. Writing each matrix row by row, with a semicolon between rows:

The top left entry is 3 + 1 = 4, and so on. Subtraction works the same way. Matrices of different sizes can't be added.

Example 2: multiplying a 2 by 3 and a 3 by 2

To multiply, the number of columns in A must equal the number of rows in B. A 2 by 3 times a 3 by 2 works, and the answer is 2 by 2. Each entry is a row of A times a column of B: multiply matching entries and add.

Take A = [2, 0, 1; 1, 3, −1], a 2 by 3, and B = [1, 2; 0, 1; 4, −1], a 3 by 2:

So AB = [6, 3; −3, 6]. The calculator shows a line like these for each entry. The matrix multiplication calculator guide goes through the method in more detail.

Example 3: determinant of a 3 by 3 matrix

Choose Determinant and set A to 3 by 3. The calculator expands along the top row, with signs plus, minus, plus. Each entry multiplies the 2 by 2 determinant left when you cover its row and column.

The determinant is −9. A 2 by 2 determinant is ad − bc. For 4 by 4 and 5 by 5, the calculator row reduces to a triangle and multiplies the diagonal, which is much quicker than cofactors.

Example 4: inverse of a 2 by 2 matrix

For a 2 by 2 matrix, swap the two entries on the main diagonal, change the signs of the other two, and divide by the determinant. For A = [4, 7; 2, 6], det A = 4 × 6 − 7 × 2 = 10, so:

The calculator gives the fractions exactly. For 3 by 3 and bigger it uses Gauss-Jordan elimination, which the inverse matrix calculator guide works through. If the determinant is 0, there is no inverse.

Example 5: solving simultaneous equations with RREF

To solve x + 2y = 5 and 3x − y = 1, write the augmented matrix, with the numbers on the right as a last column, and choose Row reduce:

The calculator shows each row operation like this. Reading the reduced rows, x = 1 and y = 2. The rank is 2. A rank lower than the number of unknowns means there is no unique solution. For equations typed as equations, the simultaneous equations calculator is quicker.

Example 6: eigenvalues and eigenvectors

The calculator opens on A = [2, 1; 1, 2]. An eigenvector keeps its direction when A acts on it, and the eigenvalue is how much it is stretched. For a 2 by 2 matrix the characteristic polynomial is λ² − (trace)λ + det A. The trace is 2 + 2 = 4 and det A = 3:

The eigenvalues are 1 and 3, with eigenvectors (1, −1) and (1, 1). Check: A times (1, 1) is (3, 3), which is 3 times (1, 1). When the polynomial has no simple roots the calculator gives decimals, and it says when eigenvalues are complex.

Example 7: a 2 by 2 transformation

Choose 2×2 transformation and type A = [0, −1; 1, 0]. The calculator draws the grid before and after and names it: a rotation by 90 degrees anticlockwise. It also gives the area scale factor, which is the size of the determinant. The point (3, 1) goes to (−1, 3). It recognises rotations, reflections, enlargements, stretches and shears.

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Reading the answers

• Determinant 0 means the matrix squashes space flat: it has no inverse, and its rows are not independent.
• The sign of the determinant says whether a transformation flips the plane over (negative) or not (positive).
• Rank is the number of independent rows. For an n by n matrix, rank n means it has an inverse.
• Order matters in multiplication. AB and BA are usually different, and one of them may not exist at all.
• The identity matrix I, 1s down the diagonal and 0s elsewhere, acts like the number 1: AI = IA = A.

Common mistakes

• Multiplying entry by entry. Matrix multiplication is rows times columns, not matching positions.
• Checking the wrong sizes. For A × B, the inside numbers must match: (2 by 3)(3 by 2) works, (2 by 3)(2 by 3) doesn't.
• Swapping the order of a product. A × B and B × A are different calculations.
• Missing the minus sign in the middle term of a 3 by 3 determinant: the signs go plus, minus, plus.
• Dividing by a zero determinant. If det A = 0, there is no inverse, so stop.
• Rounding fractions to decimals part way through row reduction. Keep them as fractions, as the calculator does.

Read next

Maths · AlgebraInverse matrix calculator: 2x2, 3x3 and up with stepsFind the inverse of a matrix up to 5 by 5 with exact fractions: the 2 by 2 formula, Gauss-Jordan steps for 3 by 3 and bigger, and examples to check.