Matrix Calculator with Exact Fractions and Row-Reduction Steps

How to use the matrix calculator

1. Set the rows and columns of A with the two drop-downs (1 to 6 each), then type the entries. Use B for A + B, A - B, A × B and the right-hand side of a system.
2. Press an operation: det(A), A⁻¹, RREF(A), transpose, rank, eigenvalues, Aⁿ (set n from -20 to 50), or Solve A·x = B.
3. For RREF, the inverse and Solve, open the "Row operations" list under the result. Every step is shown with the matrix after it.

What RREF, rank, determinant, inverse and eigenvalues tell you

Row reduction rewrites a matrix using three legal moves: swap two rows, multiply a row by a nonzero number, add a multiple of one row to another. These moves never change the solutions of a system. RREF (reduced row echelon form) is the end of the road: each pivot is 1, and it is the only nonzero entry in its column.

The rank is the number of pivots. It counts how many equations carry independent information. The determinant of a square matrix is one number, and it is zero exactly when the rows are dependent. For a 2 × 2 matrix it is ad - bc. The inverse A⁻¹ satisfies A·A⁻¹ = I and exists only when the determinant is not zero. The tool finds it by row-reducing [A | I] until the left half becomes I.

An eigenvalue λ of A is a number for which A·v = λ·v has a nonzero solution v. Eigenvalues are the roots of det(A - λI) = 0. Their sum is the trace (the diagonal total) and their product is the determinant, which is a handy check.

One, none or infinitely many solutions

Row-reduce the augmented matrix [A | b] and read the pivots. Three examples, all from the tool:

A pivot in every unknown column and none in the b column means exactly one solution. A pivot in the b column means none. Fewer pivots than unknowns, with a consistent b, means infinitely many.

Three worked examples

1. Mesh currents in a three-loop circuit, solved by RREF

Take clockwise currents I₁, I₂, I₃ in amps. Loop 1 has 6 Ω of its own, shares 2 Ω with loop 2, and has a 12 V source. Loop 2 has 4 Ω of its own and shares 3 Ω with loop 3. Loop 3 has 5 Ω of its own and a 6 V source working against its current. Kirchhoff's voltage law gives:

8·I₁ - 2·I₂ = 12
-2·I₁ + 9·I₂ - 3·I₃ = 0
-3·I₂ + 8·I₃ = -6

Enter the coefficients in A, the column 12, 0, -6 in B, and press Solve. The determinant of A is 472, so the answer is unique. The tool's 8 row operations:

1. R1 → R1 ÷ 8, giving [1, -1/4, 0 | 3/2]
2. R2 → R2 + 2·R1, giving [0, 17/2, -3 | 3]
3. R2 → R2 ÷ (17/2), giving [0, 1, -6/17 | 6/17]
4. R1 → R1 + (1/4)·R2, giving [1, 0, -3/34 | 27/17]
5. R3 → R3 + 3·R2, giving [0, 0, 118/17 | -84/17]
6. R3 → R3 ÷ (118/17), giving [0, 0, 1 | -42/59]
7. R1 → R1 + (3/34)·R3, giving [1, 0, 0 | 90/59]
8. R2 → R2 + (6/17)·R3, giving [0, 1, 0 | 6/59]

So I₁ = 90/59 ≈ 1.5254 A, I₂ = 6/59 ≈ 0.1017 A and I₃ = -42/59 ≈ -0.7119 A. The negative sign means loop 3 actually runs counterclockwise. Substituting the fractions back gives exactly 12, 0 and -6. For more on this method see simultaneous equations with matrices.

Load this example

2. A 3 × 3 inverse with fractions

Let A = [[2, 1, 0], [1, 3, 1], [0, 1, 2]]. This tridiagonal pattern turns up in chains of springs and ladder networks. By hand, det(A) = 2(6 - 1) - 1(2 - 0) = 8, which the tool confirms. Row-reducing [A | I] takes 8 operations; the first three are R1 ÷ 2, R2 - R1, and R2 ÷ (5/2). The result is:

A⁻¹ = [[5/8, -1/4, 1/8], [-1/4, 1/2, -1/4], [1/8, -1/4, 5/8]]

Multiplying A by this matrix returns the identity exactly. Now A·x = (1, 2, 3) is one multiplication: x = A⁻¹·b = (1/2, 0, 3/2). The numeric eigenvalues of A are 1, 2 and 4, and their product is 8, matching the determinant.

Load this example

3. Eigenvalues of a symmetric 2 × 2 stress matrix

A plane stress state in MPa is S = [[60, 20], [20, 40]], with 60 and 40 as normal stresses and 20 as shear. The characteristic equation is λ² - 100λ + 2000 = 0, since the trace is 100 and the determinant is 2000. The tool returns:

• λ = 72.36068 with unit eigenvector (0.850651, 0.525731)
• λ = 27.63932 with unit eigenvector (-0.525731, 0.850651)

These are the principal stresses, 50 ± 10√5 MPa. The eigenvectors give their directions: the larger one sits 31.72° from the first axis, and the two are perpendicular. This is not luck. A real symmetric matrix always has real eigenvalues and perpendicular eigenvectors.

Load this example

Reading the result and common mistakes

Why exact fractions matter. The matrix [[1, 1/3], [1/3, 1]] has determinant 8/9 and inverse [[9/8, -3/8], [-3/8, 9/8]]. The tool keeps 1/3 as 1/3 through every step, so A times that inverse is exactly the identity. Rounded decimals can also hide a zero: a pivot that should be 0 may come out as a tiny nonzero number, and the rank is then wrong. Fractions remove that risk.

• Singular matrix. [[1,2,3],[4,5,6],[7,8,9]] has determinant 0 and rank 2, so it has no inverse. The tool reports this plainly.
• Wrong shape. Determinant, inverse, eigenvalues and powers need a square matrix. A × B needs the columns of A to equal the rows of B, and A × B is usually not B × A.
• Irrational entries. One sqrt(2) turns the whole run into decimals, so you lose exact fractions.
• Eigenvalues are numeric. They are decimals to about 10 digits, and complex eigenvalues are shown as a + bi without eigenvectors.

Frequently asked questions

Related tools

ToolScientific CalculatorCasio-style calculator with 8 modes: fractions, equations, matrices, complex numbers, base-N and statistics. ToolStatistics CalculatorMean, SD, quartiles, outliers, regression with R², and normal, t, chi-square and F distributions. ToolGraphing CalculatorPlot functions, implicit curves, inequalities, polar and parametric graphs, with sliders. ToolUnit Converter19 engineering categories plus a free-form box for compound units like 9.81 m/s^2 to ft/s^2.