How do I multiply two matrices?

Select the 'A × B' operation, enter values for Matrix A and Matrix B, and the result appears instantly. For multiplication, each cell (i,j) of the result is the dot product of row i from A and column j from B. Note: for NxN square matrices, A × B generally does not equal B × A.

What is a matrix determinant and what does it mean?

The determinant is a scalar value that encodes key properties of a matrix. A non-zero determinant means the matrix is invertible (non-singular). A determinant of zero means the matrix is singular — it has no inverse, and the rows are linearly dependent. Geometrically, the absolute value of the determinant represents the scaling factor of the transformation the matrix represents.

How do I find the inverse of a matrix?

Select the 'A⁻¹' operation and enter your matrix. The inverse only exists when det(A) ≠ 0. For a 2×2 matrix [[a,b],[c,d]], the inverse is (1/(ad−bc)) × [[d,−b],[−c,a]]. This calculator uses Gauss-Jordan elimination for 3×3 and 4×4 matrices.

What is RREF (Row Reduced Echelon Form)?

RREF is a standardized form of a matrix where: (1) the leading entry of each non-zero row is 1, (2) each leading 1 is the only non-zero entry in its column, and (3) rows are ordered top to bottom. It's used to solve systems of linear equations, find the rank, and determine if a system has no solution, one solution, or infinitely many.

What is the rank of a matrix?

The rank is the number of linearly independent rows (or equivalently, columns) in the matrix. It equals the number of non-zero rows in the RREF. For an N×N matrix, full rank (rank = N) means the matrix is invertible. A rank less than N indicates linear dependence among rows.

Why does the matrix inverse show 'singular matrix' error?

A matrix is singular when its determinant is exactly zero. This means its rows (or columns) are linearly dependent — one row can be expressed as a combination of the others. Singular matrices cannot be inverted. For example, any matrix with a row of all zeros, or two identical rows, will be singular.

Why are eigenvalues not supported for 4×4 matrices?

Finding eigenvalues of a 4×4 matrix requires solving a degree-4 polynomial (the characteristic polynomial). While formulas exist, they are extremely complex and numerically unstable in floating-point arithmetic. For 2×2 and 3×3 matrices, this calculator uses the quadratic and Cardano's cubic formulas respectively, which are numerically precise for typical inputs.

Matrix Calculator

Matrix Size

Matrix A

Matrix B

A × B — 3×3 Matrix

138

114

Matrix Properties

Derived properties of the result

Trace

Sum of diagonal elements

189

What Is a Matrix?

Understanding matrices and why they're fundamental to linear algebra

A matrix is a rectangular grid of numbers arranged in rows and columns. Matrices are the core data structure of linear algebra — they represent transformations, encode systems of equations, and describe relationships between variables in machine learning, physics, and engineering.

This matrix calculator handles nine operations on 2×2, 3×3, and 4×4 square matrices. Binary operations (multiply, add, subtract) use both Matrix A and Matrix B; all other operations work on Matrix A alone. Results update as you type — no button needed.

Where matrix calculations are used

Computer graphics: Rotation, scaling & projection transforms

Machine learning: PCA, covariance matrices, neural network weights

Systems of equations: Ax = b solved via RREF or A⁻¹b

Physics & engineering: Eigenvalues for vibration modes & stability

Key Matrix Formulas

Determinant, inverse, eigenvalue, and multiplication formulas with variable definitions

2×2 Determinant

det(A) = ad − bc

a, d = main diagonal

b, c = off diagonal

2×2 Inverse

A⁻¹ = (1 / det(A)) × [[d, −b], [−c, a]]

det(A) = must be ≠ 0

I = identity matrix

Eigenvalues (2×2)

λ = (tr(A) ± √(tr(A)² − 4·det(A))) / 2

tr(A) = trace = a + d

λ = eigenvalue scalar

Matrix Multiplication

C[i][j] = Σₖ A[i][k] × B[k][j]

i, j = row / col of result

k = inner dimension index

Worked Example — 2×2 Determinant & Inverse

Given A = [[3, 1], [2, 4]]:

det(A) = (3)(4) − (1)(2) = 12 − 2 = 10

A⁻¹ = (1/10) × [[4, −1], [−2, 3]] = [[0.4, −0.1], [−0.2, 0.3]]

Verify: A × A⁻¹ = [[3(0.4)+1(−0.2), 3(−0.1)+1(0.3)], [2(0.4)+4(−0.2), 2(−0.1)+4(0.3)]] = [[1, 0], [0, 1]] = I ✓

Matrix Operations Explained

What each operation does and when to use it

A × BMultiply

Each cell (i,j) = dot product of row i from A and column j from B. A×B ≠ B×A in general.

A + BAdd

Element-wise addition. Matrices must be the same size. (A+B)[i][j] = A[i][j] + B[i][j].

A − BSubtract

Element-wise subtraction. Matrices must be the same size. (A−B)[i][j] = A[i][j] − B[i][j].

det(A)Determinant

Scalar encoding how A scales space. Zero = singular (no inverse). Computed via cofactor expansion.

A⁻¹Inverse

Matrix where A × A⁻¹ = I. Only exists when det(A) ≠ 0. Uses Gauss-Jordan elimination.

AᵀTranspose

Swap rows and columns: Aᵀ[i][j] = A[j][i]. Useful in dot products and symmetric matrices.

RREFRow Echelon

Standardized form with leading 1s in each pivot row. Used to solve Ax = b systems.

rank(A)Rank

Number of linearly independent rows. Equals the number of non-zero rows in RREF.

λEigenvalues

Scalars λ where Av = λv for non-zero v. Supported for 2×2 (quadratic) and 3×3 (Cardano).

Solving Real Problems with Matrices

Practical tips for equations, eigenvalues, and matrix powers

Solving Ax = b

To solve a square system, first check det(A) ≠ 0. If it is invertible, use A⁻¹ — the solution is x = A⁻¹b. Alternatively, apply RREF to A to confirm full rank (rank = N means a unique solution exists).

Check invertibility first

Run det(A) before computing A⁻¹. If det = 0, the inverse doesn't exist and you'll get an error.

Complex eigenvalues

If eigenvalues have an imaginary part (shown as ± xi), the matrix represents oscillatory or rotational behaviour in a dynamic system.

Matrix power

To compute A² or A³, multiply A × A successively. Eigenvalues of Aⁿ are the eigenvalues of A raised to the nth power.

Clear and Identity presets

Use the Clear button to zero out a matrix and Identity to reset it to I. Useful as a starting point for constructing custom matrices.

Common Matrix Mistakes

Pitfalls to avoid when calculating with matrices

Assuming A × B = B × A

Mistake: Treating matrix multiplication as commutative, like regular numbers

Correct: Matrix multiplication is NOT commutative — A×B and B×A are generally different. Always check the order.

Inverting a singular matrix

Mistake: Trying to compute A⁻¹ without checking the determinant first

Correct: Always check det(A) ≠ 0 before inverting. If det = 0 the matrix is singular and has no inverse.

Confusing transpose with inverse

Mistake: Using Aᵀ as a shortcut for A⁻¹ — "flipping the matrix should undo it"

Correct: Aᵀ swaps rows and columns; A⁻¹ is the unique matrix where A×A⁻¹ = I. They are equal only for orthogonal matrices.

Reading RREF wrong

Mistake: Assuming every RREF is an identity matrix

Correct: RREF may have zero rows (indicating rank deficiency). Count the non-zero rows to find the rank.

Ignoring complex eigenvalues

Mistake: Discarding eigenvalues with an imaginary part as errors

Correct: Complex eigenvalues are real results — they indicate rotation or oscillation. They always come in conjugate pairs (a+bi, a−bi).

Matrix Size Reference

Properties by matrix dimension

Property	2×2	3×3	4×4
Elements	4	9	16
Max rank	2	3	4
Eigenvalues	Quadratic	Cubic	Not supported
Det method	ad−bc	Cofactor	Cofactor
Inverse method	Adjugate	Gauss-Jordan

Why stop at 4×4? Larger matrices require numerical algorithms (LU decomposition, QR iteration) that are better served by libraries like NumPy or MATLAB. This calculator uses floating-point arithmetic with cofactor expansion, Gauss-Jordan elimination, and Cardano's cubic formula for the supported sizes.

Frequently Asked Questions

Common questions and detailed answers

Eigenvalues (λ) are special scalars where Av = λv for some non-zero vector v (the eigenvector). They reveal fundamental properties: in PCA they represent variance explained by each component; in differential equations they determine stability; in physics they represent energy levels. This calculator finds eigenvalues for 2×2 matrices (quadratic formula) and 3×3 matrices (Cardano's method for the cubic characteristic polynomial).

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Last updated Apr 2, 2026