Matrix Calculator

Matrices turn up wherever several linear relationships act at once: systems of equations, rotations in computer graphics, population models that step forward one year at a time. The operations are simple to define but error-prone by hand, since a single 3 × 3 product already involves 27 multiplications. The calculator does the arithmetic exactly and shows the intermediate work, so you can check homework line by line.

The page links to three sister tools. The RREF Calculator lists every row operation of Gauss-Jordan elimination on any matrix. The System of Equations Calculator takes equations written out in x, y and z. The Vector Calculator handles dot products, cross products and angles. A note on notation: in the text below a matrix is written row by row, with commas between entries and semicolons between rows, so [1, 2; 3, 4] has first row 1, 2 and second row 3, 4. You can paste that exact text into the calculator.

Matrix sizes and which operations are allowed

A matrix with m rows and n columns is called an m × n matrix, always rows first. Adding two matrices pairs up matching entries, so they must be the same size. Multiplying pairs rows of the first with columns of the second, so the inner dimensions must agree. The determinant, the trace, the inverse, powers and eigenvalues are only defined for square matrices. The table summarizes the rules the calculator checks before it computes anything.

The formulas behind each operation

Each operation is a definition rather than an approximation, which is why exact answers are possible:

Here M_ij is the minor of entry a_ij, the determinant of the smaller matrix left after deleting row i and column j, and (−1)^i+jM_ij is its cofactor. The determinant of a product is the product of the determinants, det(AB) = det(A)·det(B), and an inverse exists exactly when det(A) ≠ 0. The eigenvalues satisfy two identities that make good checks: they add up to the trace and multiply to the determinant.

Worked example 1: a 2 × 3 matrix times a 3 × 2 matrix

Take A = [2, −1, 3; 0, 4, 1], a 2 × 3 matrix, and B = [1, 2; 3, 0; −2, 5], a 3 × 2 matrix. The inner dimensions match (3 and 3), so AB exists and has the outer dimensions, 2 × 2. Each entry is a row of A dotted with a column of B:

So AB = [−7, 19; 10, 5], with determinant (−7)(5) − (19)(10) = −225. Now press Swap A and B and multiply again. BA is a completely different object: a 3 × 3 matrix, [2, 7, 5; 6, −3, 9; −4, 22, −1]. Its determinant is 0, because a product that passes through a 2-dimensional middle step can have rank at most 2. One quantity survives the swap, though: the trace. tr(AB) = −7 + 5 = −2 and tr(BA) = 2 − 3 − 1 = −2. That identity, tr(AB) = tr(BA), holds for any sizes where both products exist. The transpose of the product reverses the order as well: (AB)^T = B^TA^T = [−7, 10; 19, 5].

Worked example 2: a 3 × 3 determinant and the exact inverse

Let A = [1, 2, −1; 3, 0, 4; 2, 1, 1], the matrix the calculator loads by default. Expanding along row 1, the signs run +, −, +, and each entry multiplies the 2 × 2 determinant left after crossing out its row and column:

Any row or column gives the same value, and a line with a zero saves work. Column 2 holds 2, 0 and 1 with signs −, +, −, so det(A) = −2·(3·1 − 4·2) + 0 − 1·(1·4 − (−1)·3) = 10 + 0 − 7 = 3. Pick Column 2 under Cofactor expansion along to see this version in full.

Because det(A) = 3 is not zero, A is invertible. Gauss-Jordan elimination on [A | I] takes eight row operations and produces

Every entry is a multiple of 1/3, which is no accident: A⁻¹ equals the adjugate (the transposed matrix of cofactors) divided by det(A), and here the adjugate is the integer matrix [−4, −3, 8; 5, 3, −7; 3, 3, −6]. In decimal form the inverse reads −1.3333, −1, 2.6667 in the first row, and multiplying A by those rounded numbers no longer returns the identity exactly. The fractions do. Two further checks follow from the determinant rules: det(A⁻¹) = 1/3, and doubling every entry of this 3 × 3 matrix multiplies the determinant by 2³, giving det(2A) = 24.

Worked example 3: eigenvalues and eigenvectors, real and complex

An eigenvector of A is a nonzero vector that A only stretches: Av = λv. For A = [5, 4; 1, 2] the characteristic polynomial is det(λI − A) = λ² − 7λ + 6 = (λ − 6)(λ − 1), so the eigenvalues are 6 and 1. They add to 7, the trace, and multiply to 6, the determinant. For λ = 6, the system (A − 6I)v = 0 reads −x + 4y = 0, so v = (4, 1); for λ = 1 it reads 4x + 4y = 0, so v = (−1, 1). The calculator shows each eigenvector with its free variable set to 1, plus a whole-number version whenever fractions appear.

Eigenvalues predict powers. Since A = P D P⁻¹ with D = diag(6, 1), every power is Aⁿ = P Dⁿ P⁻¹, and the entries of A⁵ must grow like 6⁵ = 7,776. Run the power operation with n = 5 and you get A⁵ = [6221, 6220; 1555, 1556], which matches the closed form (4·7,776 + 1)/5 = 6,221 exactly.

Not every real matrix has real eigenvalues. For [1, −2; 1, 3] the polynomial is λ² − 4λ + 5, whose discriminant 16 − 20 = −4 is negative, so the eigenvalues are the conjugate pair 2 + i and 2 − i, shown exactly as 2 ± i. The eigenvector for 2 + i is (−1 + i, 1). The 3 × 3 preset [4, −1, −1; 0, 1, 0; −1, 3, 4] has characteristic polynomial λ³ − 9λ² + 23λ − 15, eigenvalues 5, 3 and 1, and eigenvectors (−1, 0, 1), (1, 0, 1) and (0, −1, 1).

Powers of the Fibonacci matrix

The matrix F = [1, 1; 1, 0] produces the Fibonacci numbers when you raise it to a power: Fⁿ = [F_n+1, F_n; F_n, F_n−1]. Because det(F) = −1, the product rule gives det(Fⁿ) = (−1)ⁿ, which is Cassini's identity F_n+1F_n−1 − F_n² = (−1)ⁿ. The trace of Fⁿ runs through the Lucas numbers. Every value below was computed with the calculator's exact power routine (load the Fibonacci power preset and change n).

Negative powers work too, because det(F) ≠ 0: F⁻¹ = [0, 1; 1, −1] and F⁻³ = [−1, 2; 2, −3], which extends the Fibonacci sequence backward with alternating signs.

Singular matrices, rank and why some inverses do not exist

A square matrix with determinant 0 is called singular. The classic example is [1, 2, 3; 4, 5, 6; 7, 8, 9]: the middle row is the average of the other two, so the rows are linearly dependent. Its determinant is 0, its reduced row echelon form is [1, 0, −1; 0, 1, 2; 0, 0, 0] and its rank is 2, one short of full rank, with nullity 3 − 2 = 1. The Singular matrix preset shows the calculator's response: Gauss-Jordan elimination runs out of pivots in the third column, so no inverse exists. For any n × n matrix these statements are equivalent: det(A) ≠ 0, rank(A) = n, the reduced form is the identity, Ax = b has exactly one solution for every b, and 0 is not an eigenvalue.

Solving Ax = b with the augmented matrix

The three equations x + 2y − z = −3, 3x + 4z = 18 and 2x + y + z = 6 have coefficient matrix A = [1, 2, −1; 3, 0, 4; 2, 1, 1] and right-hand side b = (−3, 18, 6). Choose Solve Ax = b and the calculator writes the augmented matrix [A | b], row-reduces it in eight operations and reads off x = 2, y = −1, z = 3. If the reduced matrix contains a row 0 = 1, the system has no solution; if some variables have no pivot, they become free parameters and the solution is written in parametric form, such as x = (0, 3, 0) + t(1, −2, 1). For long systems, or to see every intermediate matrix with explanations, the RREF Calculator goes into more detail.

How many multiplications a determinant needs

Cofactor expansion is the method taught first, and it is the one this calculator shows for 3 × 3 matrices. It does not scale. Each n × n determinant expands into n smaller ones, so the work grows like n factorial, while Gaussian elimination grows only like n³. The table counts multiplications and divisions for a full matrix, without additions; the elimination column assumes no zero pivots.

For 3 × 3 the two methods cost about the same (9 versus 10 operations), which is why hand calculation uses cofactors. From 4 × 4 on, elimination wins, and at 10 × 10 cofactor expansion needs more than eighteen thousand times as many multiplications. The calculator therefore switches to elimination from 4 × 4: a row swap flips the sign of the determinant, and adding a multiple of one row to another leaves it unchanged.

How to use this calculator, and who it is for

Pick an operation first, because the inputs adapt to it: matrix B appears only for A + B, A − B and A × B, and fields for k, n or b appear when an operation needs them. Set the size of each matrix with the menus (up to 6 × 6) and type the entries: integers, decimals such as 0.25 and fractions such as 3/4 all work, and empty cells count as 0. Paste as text accepts a whole matrix at once, one row per line or rows separated by semicolons; write numbers without thousands separators (1000, not 1,000). The Try buttons load the main worked examples on this page. In decimal mode, a nonzero value too small for the chosen number of decimal places is shown in e-notation, such as 3.333e-5, rather than as 0. Results update as you type, and Use this result as A chains operations, for example squaring a matrix and then taking its determinant.

Students in linear algebra, precalculus and engineering math use it to check matrix products, cofactor expansions, inverses and eigenvalue problems against their own work. Teachers use the presets to build examples with clean answers. Programmers and data analysts sanity-check small transformation, Markov or covariance matrices before trusting code that handles large ones, and engineers solve small systems from circuit and statics problems.

Key terms

• Identity matrix I: ones on the main diagonal and zeros elsewhere; AI = IA = A.
• Determinant: a single number computed from a square matrix; zero exactly when the matrix is singular.
• Minor and cofactor: the determinant left after deleting row i and column j, and that minor times (−1)^i+j.
• Inverse A⁻¹: the matrix with AA⁻¹ = A⁻¹A = I; it exists only when det(A) ≠ 0.
• Rank: the number of linearly independent rows (equivalently columns), equal to the number of pivots after row reduction.
• Eigenvalue and eigenvector: a number λ and a nonzero vector v with Av = λv.
• Algebraic and geometric multiplicity: how often an eigenvalue repeats as a root of the characteristic polynomial, and how many independent eigenvectors it has. When the second falls short of the first, the matrix is not diagonalizable.
• Augmented matrix: the coefficient matrix with the right-hand side attached as an extra column, written [A | b].

Limitations and numerical notes

Exact arithmetic covers everything except eigenvalues that are irrational or complex. Those come from a numerical routine and are rounded for display, so an eigenvalue such as (5 + √33)/2 for [1, 2; 3, 4] appears as 5.3723 at four decimal places; for 2 × 2 matrices the calculator also prints the exact radical form. Rational eigenvalues are snapped to fractions only after they are confirmed to be exact roots of the characteristic polynomial, so a printed 6 really is 6. Multiplicities are exact as well: dividing the characteristic polynomial by its greatest common factor with its derivative leaves one copy of each root, which tells the calculator how many distinct eigenvalues there are, so rounding can never split a repeated eigenvalue or merge two close ones. Exact entries can grow quickly under powers, so Aⁿ is shown only while every entry has at most 300 digits. The 6 × 6 limit keeps the step-by-step output readable, and entries must be numbers, so a matrix containing a variable or a parameter k needs a computer algebra system.

Related calculators

• To see every row operation that reduces a matrix, use the RREF Calculator.
• To solve two or three equations typed in x, y and z, use the System of Equations Calculator.
• For dot products, cross products, magnitudes and angles, use the Vector Calculator.
• For arithmetic with complex eigenvalues such as 2 + i, use the Complex Number Calculator.
• The eigenvalues of a 2 × 2 matrix are the roots of a quadratic; the Quadratic Formula Calculator solves one directly.
• To simplify or combine the fractions in an exact inverse, use the Fraction Calculator, and for more Fibonacci numbers use the Fibonacci Calculator.

Sources

Matrix operations are exact mathematical definitions, so the worked examples and tables above were computed directly with the formulas shown. The definitions and identities follow these references:

• NIST Digital Library of Mathematical Functions, §1.2 Elementary Algebra: matrix multiplication (defined only when the inner dimensions agree, not commutative), the transpose, the determinant, the inverse (unique when det(A) ≠ 0), the trace, and eigenvalues as roots of det(A − λI) = 0, with the determinant equal to their product and the trace equal to their sum (§1.2(v) and §1.2(vi)).
• NIST Digital Library of Mathematical Functions, §1.3 Determinants: minors, cofactors, expansion along a row, the effect of row swaps and row additions, and det(AB) = det(A)det(B).
• MIT OpenCourseWare 18.06SC Linear Algebra: Multiplication and Inverse Matrices: matrix multiplication from several points of view, including row times column, and Gauss-Jordan elimination for the inverse.
• MIT OpenCourseWare 18.06SC: Determinant Formulas and Cofactors and Eigenvalues and Eigenvectors, and Diagonalization and Powers of A.
• math.js documentation, function eigs: the numerical eigenvalue routine used for eigenvalues that are not rational.