Matrix Calculator

Written byOleh Soloninupdated 27 September 2026

What to do

Matrix A

One matrix row per line, numbers separated by spaces. Fractions (1/3) and decimals (0.5) work.

Result

Size2×2
Determinant10
Inverse matrix
3/5 −7/10 −1/5 2/5
Invertibleyes, the determinant is not zero
Rank2
Trace10
Transpose
4 2 7 6
Workingdet = 4 × 6 − 7 × 2 = 10; A⁻¹ = 1/10 × [6 −7; −2 4]

Exact fractions instead of rounded decimals

Finds the inverse matrix in exact fractions, the determinant, rank, trace and transpose, multiplies, adds and subtracts matrices up to 6×6, and shows results as tables.

An inverse exists only for a square matrix with a non-zero determinant. For A = [4 7; 2 6] the determinant is 4 × 6 − 7 × 2 = 10, and A⁻¹ = 1/10 × [6 −7; −2 4] = [3/5 −7/10; −1/5 2/5]. The calculator writes it in fractions, as an answer to an exercise would.

Matrices can be multiplied when the first has as many columns as the second has rows: a 2×3 times a 3×2 gives a 2×2. Adding and subtracting work only on matrices of the same size, entry by entry.

How to use

Choose the operation: inverse and determinant, multiply A × B, add or subtract.
Enter the matrix row by row: one row per line, numbers separated by spaces. Fractions (1/3) and decimals (0.5) work. For multiplying or adding, enter matrix B as well.
Read the result as a table; for a 2×2 matrix, the formula working too.

Finding the inverse

For a 2×2 matrix [a b; c d] the inverse is 1/(ad − bc) × [d −b; −c a]: swap the main-diagonal entries, negate the other two, and divide by the determinant.

For anything larger the calculator uses Gauss–Jordan elimination: place the identity matrix beside yours and reduce the left side to the identity with row operations; the right side becomes the inverse. The determinant and the rank, the number of non-zero rows after reduction, come out along the way.

Worked out with this calculator.
Matrix	Determinant	Inverse
[4 7; 2 6]	10	[3/5 −7/10; −1/5 2/5]
[2 1; 1 1]	1	[1 −1; −1 2]
[1 2; 2 4]	0	none: the rows are proportional
[2 0 1; 1 3 2; 1 1 1]	0	none, rank 2

Multiplying matrices

The entry of C = A × B in row i and column j is the sum of the products of row i of A with column j of B. For A = [1 2 3; 4 5 6] and B = [7 8; 9 10; 11 12], c₁₁ = 1 × 7 + 2 × 9 + 3 × 11 = 58, and the product is [58 64; 139 154].

Matrix multiplication is not commutative: A × B and B × A usually differ, and one of them may not even be defined because of the sizes.

What this calculator shows

The inverse in exact fractions, when it exists
Determinant, rank and trace
The transpose as a table
The product, sum or difference of two matrices
The working for a 2×2 matrix and for the first entry of a product

What to keep in mind

Matrices up to 6×6. Everything is computed in exact fractions, with no rounding.
Determinant, trace and inverse apply to square matrices only; rank and transpose to any.

FAQs

How do I find the inverse of a 2×2 matrix?

Swap the main-diagonal entries, change the sign of the other two and divide everything by the determinant: for [4 7; 2 6] that is 1/10 × [6 −7; −2 4].

When does a matrix have no inverse?

When it is not square or its determinant is zero, meaning its rows or columns are linearly dependent and its rank is below its size.

How do I multiply two matrices?

Each entry of the result is a row of the first matrix multiplied into a column of the second. The first matrix needs as many columns as the second has rows.

What is the rank of a matrix?

The largest number of linearly independent rows or columns, which after reduction to echelon form is the number of non-zero rows.

What is a transpose?

Rows turned into columns: the first row becomes the first column. The transpose of a 2×3 matrix is 3×2.

What does a determinant of zero mean?

The matrix has no inverse, and the system it represents has either no solution or infinitely many. Geometrically the transformation squeezes space into fewer dimensions — as many as the rank: a plane onto a line or a point, three-dimensional space onto a plane, a line or a point.

Why is matrix multiplication not commutative?

Because it composes transformations, and order changes the outcome: rotating then stretching is not the same as stretching then rotating. AB and BA are generally different, and one may not even be defined.

Worked example

Inverse of a 2×2 matrix

Input: 4 7 / 2 6

Output: Determinant 10, inverse [3/5 −7/10; −1/5 2/5]

Note: A⁻¹ = 1/10 × [6 −7; −2 4]. Check: A × A⁻¹ is the identity matrix.

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Result

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