A matrix is a rectangular array of numbers arranged in rows and columns. Matrices are fundamental in linear algebra and are used to represent systems of equations, transformations, and data in fields like physics, economics, and machine learning.

How do you multiply two matrices?

Matrix multiplication requires the number of columns in the first matrix to equal the number of rows in the second. The element at position (i,j) in the result is the dot product of row i from matrix A and column j from matrix B. Matrix multiplication is not commutative — A×B does not generally equal B×A.

When does a matrix have an inverse?

A matrix has an inverse only if it is square and its determinant is non-zero. If the determinant equals zero, the matrix is called singular and no inverse exists. The inverse A⁻¹ satisfies A × A⁻¹ = I, where I is the identity matrix.

What is a determinant used for?

The determinant of a square matrix tells you whether the matrix is invertible (det ≠ 0), encodes the scale factor of the linear transformation it represents, and is used in Cramer's Rule to solve systems of equations. A zero determinant means the rows or columns are linearly dependent.

What is the rank of a matrix?

The rank is the number of linearly independent rows (or columns) in a matrix. It equals the number of non-zero rows in the row echelon form. An n×n matrix has full rank (rank = n) if and only if its determinant is non-zero and it is invertible.

What row operations are used in Gaussian elimination?

Three elementary row operations are used: (1) Swap two rows, (2) Multiply a row by a non-zero scalar, (3) Add a multiple of one row to another. These preserve the solution set of the system while transforming the matrix into row echelon form.

What is the difference between Gaussian elimination and Cramer's Rule?

Both solve systems of linear equations. Gaussian elimination works for any size system and handles all cases (unique, infinite, or no solution). Cramer's Rule uses determinants and only works when the coefficient matrix is square with non-zero determinant, giving a direct formula for each variable.

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Q: What are eigenvalues?

Eigenvalues λ are scalars satisfying det(A − λI) = 0 for a square matrix A. They describe how the matrix scales vectors along special directions called eigenvectors. For 2×2 matrices this leads to a quadratic equation. Eigenvalues can be real or complex.

1

Enter your matrix values

Click any cell in Matrix A or B and type a number. Use Tab to move between cells. Adjust the row/column selectors to resize the matrix (up to 4×4). Use preset buttons to quickly load common matrices like the identity matrix.

2

Select an operation

Choose from A+B, A−B, A×B, scalar multiply (k·A), transpose (Aᵀ), determinant, or inverse. Binary operations like addition require both matrices to have compatible dimensions. Unary operations like determinant only need Matrix A.

3

Read the results and steps

The result appears instantly as a formatted matrix. Scroll down to see the full step-by-step breakdown showing each row operation. Use the Properties Analyzer tab to examine matrix properties, or the System Solver tab to solve Ax = b.

C_ij = Σ A_ik·B_kj (Multiplication)

det(A) = ad − bc (2×2)

det(A) = a(ei−fh) − b(di−fg) + c(dh−eg) (3×3)

A⁻¹ = adj(A) / det(A)

λ² − tr(A)·λ + det(A) = 0 (Eigenvalues 2×2)

(Aᵀ)_ij = A_ji (Transpose)

For multiplication, if A is m×n and B is n×p, the result C is m×p. The inner dimensions must match.

For the inverse, the adjugate (adj) is the transpose of the cofactor matrix. The inverse only exists when det(A) ≠ 0.

Eigenvalues of a 2×2 matrix satisfy the characteristic equation λ² − trace(A)·λ + det(A) = 0, solved with the quadratic formula.

Determinant

A scalar value computed from a square matrix that encodes whether the matrix is invertible and represents the scale factor of the corresponding linear transformation.

Transpose

A matrix formed by swapping rows and columns. If A is m×n, its transpose Aᵀ is n×m. (Aᵀ)_ij = A_ji.

Inverse

Matrix A⁻¹ satisfying A·A⁻¹ = I (identity). Only exists for square matrices with non-zero determinant. Found by Gauss-Jordan row reduction.

Singular Matrix

A square matrix with determinant = 0. It is not invertible and represents a transformation that collapses space onto a lower dimension.

Rank

The number of linearly independent rows (or columns) in a matrix, equal to the number of non-zero rows in row echelon form. Full rank for n×n means rank = n.

Identity Matrix

An n×n matrix with 1s on the main diagonal and 0s elsewhere. Multiplying any matrix by the identity returns the original matrix. Denoted I or Iₙ.

Eigenvalue

A scalar λ such that Av = λv for some non-zero vector v. Eigenvalues describe the stretching/shrinking factors of a matrix along its eigenvector directions.

Row Echelon Form

A matrix form achieved by Gaussian elimination where each row's leading entry (pivot) is to the right of the row above's pivot. Used to compute rank and solve systems.

Augmented Matrix

A matrix [A|b] formed by appending the constants column b to the coefficient matrix A. Used to represent and solve a system of linear equations Ax = b.

Trace

The sum of all diagonal elements of a square matrix. Equal to the sum of eigenvalues. Denoted tr(A).

Example 1: 2×2 Matrix Multiplication

Multiply A = [[1,2],[3,4]] by B = [[5,6],[7,8]].

C₁₁ = 1·5 + 2·7 = 5 + 14 = 19

C₁₂ = 1·6 + 2·8 = 6 + 16 = 22

C₂₁ = 3·5 + 4·7 = 15 + 28 = 43

C₂₂ = 3·6 + 4·8 = 18 + 32 = 50

Result: [[19, 22], [43, 50]]

Example 2: 3×3 Determinant

Find det(A) for A = [[1,2,3],[4,5,6],[7,8,9]] by cofactor expansion along row 1.

det(A) = 1·(5·9 − 6·8) − 2·(4·9 − 6·7) + 3·(4·8 − 5·7)

= 1·(45 − 48) − 2·(36 − 42) + 3·(32 − 35)

= 1·(−3) − 2·(−6) + 3·(−3)

= −3 + 12 − 9

det(A) = 0 (singular matrix — rows are linearly dependent)

Example 3: Solving a 2×2 System

Solve: 2x + y = 5 and x + 3y = 10 using Gaussian elimination.

Augmented matrix: [[2, 1, | 5], [1, 3, | 10]]

R2 → R2 − (1/2)R1: [[2, 1, | 5], [0, 2.5, | 7.5]]

Back-substitute: y = 7.5 / 2.5 = 3

2x + 3 = 5 → x = 1

Solution: x = 1, y = 3

Computer Graphics & Gaming

Every 3D object you see on screen is transformed by matrices. Rotation, scaling, translation, and projection are all represented as matrix multiplications. Game engines chain these transformations together to move characters, cameras, and environments in real time. A single render frame may involve millions of matrix operations.

Machine Learning

Neural networks are fundamentally systems of matrix multiplications. Each layer computes a weighted sum of its inputs using a weight matrix, then applies a non-linear function. Training a neural network means adjusting millions of matrix entries via gradient descent. Libraries like NumPy and PyTorch are optimized specifically for fast matrix operations on CPUs and GPUs.

Economics & Operations Research

Input-output models (Leontief models) describe how industries in an economy supply each other using square matrices. Linear programming, portfolio optimization, and Markov chains all rely on matrix operations. Inverting the Leontief matrix gives the total output required to satisfy a given demand vector.

Physics & Engineering

Quantum mechanics describes physical states as vectors in Hilbert space, and observables as matrices (operators). Solving systems of differential equations for structural engineering, electrical circuits, or thermodynamics involves assembling and solving large sparse matrix systems. Eigenvalues of the stiffness matrix determine resonant frequencies of structures.

What is a matrix?

A matrix is a rectangular array of numbers arranged in rows and columns. It is denoted by dimensions like "3×4" (3 rows, 4 columns). Matrices represent linear transformations, systems of equations, and data structures in mathematics and computing.

Why can't I multiply any two matrices?

Matrix multiplication requires the number of columns in the first matrix to equal the number of rows in the second. An m×n matrix can only multiply an n×p matrix, producing an m×p result. This constraint comes from the dot-product definition of each result element.

When is a matrix invertible?

A matrix is invertible (non-singular) if and only if it is square and its determinant is non-zero. Equivalently, its rank must equal its size (full rank). A singular matrix with det = 0 has linearly dependent rows or columns and cannot be inverted.

What does the determinant represent geometrically?

For a 2×2 matrix, the absolute value of the determinant equals the area of the parallelogram formed by its column vectors. For 3×3, it's the volume of the parallelepiped. A negative determinant means the transformation reverses orientation (like a reflection).

What are eigenvalues used for?

Eigenvalues describe how a matrix stretches or compresses space along special directions (eigenvectors). They are used in principal component analysis (PCA), solving differential equations, determining stability of dynamical systems, and computing page rank in web search algorithms.

What is Cramer's Rule and when does it work?

Cramer's Rule gives an explicit formula for each variable in a linear system: xᵢ = det(Aᵢ) / det(A), where Aᵢ is A with column i replaced by b. It only works when A is square with non-zero determinant. For large systems, Gaussian elimination is more computationally efficient.

Is A×B the same as B×A?

Not in general. Matrix multiplication is not commutative — A×B usually differs from B×A. They may not even have the same dimensions. This is a key difference from regular number multiplication. However, for scalar multiplication, k·A = A·k.

What does row echelon form tell me?

Row echelon form shows the structure of a matrix after Gaussian elimination. The number of non-zero rows equals the rank. If an augmented system [A|b] has a row [0 0 … 0 | c] with c≠0, the system is inconsistent (no solution). The rank determines whether a unique or infinite solution exists.

Matrix Calculator

How to Use the Matrix Calculator

Matrix Formulas

Key Terms

Worked Examples

Example 1: 2×2 Matrix Multiplication

Example 2: 3×3 Determinant

Example 3: Solving a 2×2 System

Where Matrices Are Used

Computer Graphics & Gaming

Machine Learning

Economics & Operations Research

Physics & Engineering

Frequently Asked Questions

How to Use the Matrix Calculator

Matrix Formulas

Key Terms

Worked Examples

Example 1: 2×2 Matrix Multiplication

Example 2: 3×3 Determinant

Example 3: Solving a 2×2 System

Where Matrices Are Used

Computer Graphics & Gaming

Machine Learning

Economics & Operations Research

Physics & Engineering

Frequently Asked Questions

Related Calculators