Set the size of each matrix (up to 10 × 10), fill in the values, and choose an operation. Each matrix can be transposed, raised to a power, or reduced to its determinant or inverse on its own; the buttons at the bottom combine the two matrices.
A matrix calculator performs the operations of linear algebra on rectangular arrays of numbers. This one handles every core operation you are likely to need: addition, subtraction, and multiplication of two matrices, along with the transpose, integer powers, determinant, inverse, and scalar multiplication of a single matrix. Matrices up to 10 × 10 are supported, and any result can be copied straight back into Matrix A or Matrix B so you can chain operations together without retyping.
Matrix arithmetic is tedious and error-prone by hand. Multiplying two 4 × 4 matrices requires 64 multiplications and 48 additions, and inverting one by cofactor expansion means computing seventeen separate determinants. A single sign slip invalidates the whole answer. Letting a calculator handle the mechanics frees you to concentrate on what the numbers actually mean.
A matrix is a rectangular array of numbers, symbols, or expressions arranged in rows and columns. Matrices appear throughout physics, computer graphics, probability, statistics, calculus, numerical analysis, economics, and engineering - anywhere a system involves many related quantities at once.
The dimensions of a matrix A are written m × n, meaning A has m rows and n columns. Individual values are called elements, and a specific element is identified by two subscripts: ai,j is the element in row i and column j. So a1,3 sits in the first row and third column.
Several special forms come up constantly. A square matrix has the same number of rows and columns, and only square matrices have determinants, inverses, or powers. The identity matrix has 1s down the main diagonal and 0s elsewhere, and acts like the number 1 does in ordinary multiplication: AI = IA = A. A zero matrix contains only 0s. A diagonal matrix has non-zero entries only on the main diagonal.
Addition and subtraction work element by element, which makes them the most intuitive matrix operations. To add two matrices, add each pair of corresponding entries; to subtract, subtract them.
The one strict requirement is that both matrices must have exactly the same dimensions. A 3 × 2 matrix cannot be added to a 2 × 3 matrix, because there is no correspondence between their elements. If the sizes do not match, the calculator will tell you rather than producing something meaningless.
Matrix addition is commutative (A + B = B + A) and associative ((A + B) + C = A + (B + C)), exactly as ordinary addition is.
Multiplying a matrix by a single number - a scalar - multiplies every element by that number. Doubling a matrix doubles each of its entries. Unlike matrix multiplication, scalar multiplication has no dimensional restrictions and works on any matrix of any shape. Use the × button in either panel, with the value in the adjacent box.
Multiplying two matrices is where the arithmetic stops being intuitive. The product AB is not formed by multiplying corresponding elements. Instead, the entry in row i and column j of the product is the dot product of row i of A with column j of B: multiply the paired entries and add the results.
This imposes a dimension rule that trips up almost every student at first: the number of columns in A must equal the number of rows in B. If A is m × n and B is n × p, the product AB is m × p. The inner dimensions must match and they vanish from the result; the outer dimensions survive.
Two consequences follow. First, matrix multiplication is not commutative - AB and BA are generally different matrices, and one may be defined while the other is not. Multiplying a 2 × 3 by a 3 × 4 gives a 2 × 4, but the reverse order is undefined entirely. Second, it is associative ((AB)C = A(BC)) and distributive over addition (A(B + C) = AB + AC), so those familiar rules still hold.
Because order matters, this calculator computes AB specifically. To get BA, use the A ↔ B button to swap the two matrices, then multiply again.
The transpose of a matrix, written AT, is formed by reflecting it across its main diagonal: rows become columns and columns become rows. The element at position (i, j) moves to position (j, i), so an m × n matrix transposes into an n × m matrix.
Transposes appear throughout linear algebra and statistics. A matrix equal to its own transpose is called symmetric, a property with important consequences for eigenvalues. The transpose also reverses the order of products: (AB)T = BTAT.
Raising a matrix to a power means multiplying it by itself repeatedly: A² = AA, A³ = AAA, and so on. Because matrix multiplication requires the inner dimensions to agree, only square matrices can be raised to a power.
Matrix powers are central to Markov chains, where a transition matrix raised to the n-th power gives the probabilities of moving between states after n steps. They also appear in graph theory, where powers of an adjacency matrix count walks of a given length between vertices, and in solving systems of linear recurrence relations.
The determinant is a single number computed from a square matrix that encodes a great deal about it. Geometrically, it is the factor by which the matrix scales area (in two dimensions) or volume (in three), with a negative sign indicating that orientation is flipped.
For a 2 × 2 matrix the formula is direct: det = ad − bc, where the matrix has rows [a b] and [c d]. Larger matrices are handled by cofactor expansion: pick a row or column, and for each entry multiply it by the determinant of the smaller matrix left after deleting that entry's row and column, alternating signs in a checkerboard pattern. This calculator expands along the first column, recursing until it reaches 2 × 2 blocks.
The single most important fact about determinants is this: a matrix is invertible if and only if its determinant is non-zero. A determinant of zero means the matrix is singular - its rows or columns are linearly dependent, it collapses space into a lower dimension, and no inverse exists. This is why the calculator reports "Determinant is 0. Inverse does not exist." rather than returning a spurious answer.
The inverse of a square matrix A, written A-1, is the matrix satisfying AA-1 = A-1A = I, where I is the identity. It plays the role of a reciprocal: where dividing by a number means multiplying by its reciprocal, "dividing" by a matrix means multiplying by its inverse. There is no matrix division operation as such.
The calculator computes the inverse using the adjugate method: it builds the cofactor matrix (each entry replaced by its signed minor), transposes it to form the adjugate, and divides every entry by the determinant. Formally, A-1 = adj(A) / det(A).
Inverses matter most for solving systems of linear equations. A system written in matrix form as Ax = b has the solution x = A-1b whenever A is invertible. They are equally central to computer graphics, where inverting a transformation matrix undoes a rotation, scaling, or translation, and to statistics, where the least-squares regression formula requires inverting a matrix built from the data.
Two constraints apply: the matrix must be square, and its determinant must be non-zero. Note also that inverse entries are frequently non-terminating decimals - the inverse of a matrix of whole numbers rarely has whole-number entries, so results are shown to ten significant digits.
Computer graphics. Every rotation, scaling, translation, and perspective projection is a matrix, and combining transformations is matrix multiplication. A graphics pipeline performs millions of these per second.
Systems of equations. Any system of linear equations can be written as Ax = b and solved with matrix methods - the standard approach in engineering, economics, and the physical sciences.
Machine learning. Neural networks are essentially long chains of matrix multiplications interleaved with non-linear functions, which is precisely why GPUs, built for parallel matrix arithmetic, dominate the field.
Probability and Markov chains. Transition matrices model systems moving between states, from weather patterns to page-ranking algorithms.
Statistics. Covariance matrices, regression, and principal component analysis are all built on matrix operations.
Physics and engineering. Quantum mechanics represents observables as matrices; structural engineering uses stiffness matrices; electrical engineering solves circuit networks with them.
Cryptography. Some classical ciphers, such as the Hill cipher, encrypt using matrix multiplication and decrypt using the inverse.
Assuming AB = BA. Matrix multiplication is not commutative. Order changes the answer, and sometimes changes whether the product exists at all.
Multiplying element by element. That is not matrix multiplication - it is the Hadamard product, a different operation. True matrix multiplication uses dot products of rows against columns.
Ignoring the dimension rules. Addition needs identical dimensions; multiplication needs A's column count to equal B's row count. Most errors trace back to skipping this check.
Expecting every square matrix to be invertible. Matrices with a zero determinant have no inverse, and that is a genuine property of the matrix rather than a limitation of the calculator.
Trying to take the determinant of a non-square matrix. Determinants, inverses, and powers are defined only for square matrices.
Anything from 1 × 1 up to 10 × 10, and the two matrices can be different sizes as long as the operation you choose permits it. Addition and subtraction require identical dimensions; multiplication requires A's column count to match B's row count.
Almost certainly because the inner dimensions do not agree. For AB to exist, the number of columns in A must equal the number of rows in B. A 3 × 2 times a 3 × 2 is undefined; a 3 × 2 times a 2 × 4 gives a 3 × 4.
Because the determinant is zero, making the matrix singular. Its rows or columns are linearly dependent - one is a combination of the others - so the transformation cannot be undone. This is a mathematical fact about that matrix, not a calculator limitation.
Yes - press A ↔ B to swap the two matrices, then press AB. Because matrix multiplication is not commutative, this generally gives a different result, and it is worth computing both when order is not obvious from context.
Use the Copy To A and Copy To B links beneath any result. They load the result into that panel, including resizing it, so you can feed one operation's output straight into the next.
Because the inverse divides by the determinant, and that division rarely produces whole numbers. A matrix of integers usually has an inverse full of fractions. Results are displayed to ten significant digits, which is ample for practical work while keeping the output readable.
That the matrix collapses space into a lower dimension - a plane onto a line, for instance - destroying information that cannot be recovered. The rows are linearly dependent, the matrix has no inverse, and any linear system built on it has either no solution or infinitely many.
Not directly - there is no matrix division operation. The equivalent is multiplying by the inverse. Because multiplication is not commutative, you must be careful about which side you multiply on: AB-1 and B-1A are generally different.
This Matrix Calculator is provided for educational and general informational purposes only. Results are computed in standard double-precision floating-point arithmetic and displayed to ten significant digits, so very large, very small, or nearly singular matrices may show small rounding effects inherent to numerical computation. Verify results independently for any critical application.