How do you multiply two matrices?

Matrix multiplication: C[i][j] = sum of A[i][k] × B[k][j] for all k. The number of columns in A must equal the number of rows in B. For A(m×n) × B(n×p) = C(m×p). Example: A(2×3) × B(3×2) = C(2×2).

How do you find the determinant of a matrix?

For 2×2: det([[a,b],[c,d]]) = ad − bc. For 3×3: expand along any row or column using cofactors. For larger matrices, use LU decomposition or row reduction. The determinant equals zero if and only if the matrix is singular (non-invertible).

How do you find the inverse of a matrix?

For 2×2: A⁻¹ = (1/det) × [[d,−b],[−c,a]]. For larger matrices: augment A with the identity [A|I] and row-reduce to get [I|A⁻¹]. The inverse only exists if det(A) ≠ 0.

What is the transpose of a matrix?

The transpose Aᵀ swaps rows and columns: Aᵀ[i][j] = A[j][i]. A 3×2 matrix becomes a 2×3 matrix after transposing. A symmetric matrix equals its own transpose (A = Aᵀ).

What is RREF (Reduced Row Echelon Form)?

RREF is a canonical form obtained via Gauss-Jordan elimination. A matrix is in RREF when: (1) all zero rows are at the bottom, (2) each pivot is 1, (3) each pivot is the only non-zero entry in its column. RREF is used to solve linear systems and find matrix rank.

When can two matrices be added?

Two matrices can be added only if they have the same dimensions (same number of rows AND columns). Addition is element-wise: (A+B)[i][j] = A[i][j] + B[i][j]. Unlike multiplication, matrix addition is commutative: A+B = B+A.

What is the rank of a matrix?

The rank of a matrix is the number of linearly independent rows (or columns) — equivalently, the number of pivot positions in its RREF. A full-rank n×n matrix has rank n and is invertible. Rank tells you the dimension of the column space (or row space).

Matrix Calculator

Our Matrix Calculator handles all essential linear algebra operations in one place. Enter matrices up to 4×4, then choose addition, subtraction, multiplication, determinant, inverse, transpose, scalar multiplication, or Gauss-Jordan row reduction. Every operation shows detailed step-by-step working so you can follow the method, verify your work, or learn the process from scratch.

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Matrix A

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Matrix B

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Input Values

Operation

A + B (Addition)

A − B (Subtraction)

A × B (Multiplication)

k × A (Scalar)

Aᵀ (Transpose)

det(A) (Determinant)

A⁻¹ (Inverse)

RREF (Row Reduction)

grid_on Matrix Formulas

2×2 Determinant

det = ad − bc

2×2 Inverse

A⁻¹ = (1/det)[[d,−b],[−c,a]]

Multiplication C[i][j]

Σ A[i][k] × B[k][j]

Transpose

Aᵀ[i][j] = A[j][i]

Scalar

(kA)[i][j] = k × A[i][j]

table_chart Size Compatibility

Add / Subtract Same size

Multiply A×B A cols = B rows

det / Inverse Square only

Result of m×n × n×p m×p matrix

Transpose of m×n n×m matrix

lightbulb Quick Tips

•A×B ≠ B×A — multiplication order matters
•det = 0 means singular (no inverse)
•RREF reveals rank and solution structure
•(AB)ᵀ = BᵀAᵀ (reverse order)
•Identity matrix I: A×I = A

How to Use This Calculator

Choose Matrix Size

Select dimensions for Matrix A (and B if needed): 2×2, 3×3, or up to 4×4

Select Operation

Pick from Add, Subtract, Multiply, Transpose, Determinant, Inverse, or RREF

Enter Matrix Values

Fill in each element of Matrix A (and B for binary operations)

View Result & Steps

See the result matrix and step-by-step working showing every calculation

The Formula

Matrix multiplication: C[i][j] = sum of A[i][k] × B[k][j]. For a 2×2 matrix [[a,b],[c,d]]: det = ad−bc. Inverse exists only when det ≠ 0. Transpose swaps rows and columns. Row reduction finds the equivalent RREF using elementary row operations.

C = A × B, det(A) = ad − bc, A⁻¹ = (1/det) × adj(A)

lightbulb Variables Explained

A, B Input matrices
det(A) Determinant — scalar value from a square matrix
A⁻¹ Inverse matrix — A × A⁻¹ = I (identity)
Aᵀ Transpose — rows and columns swapped
RREF Reduced Row Echelon Form via Gauss-Jordan

tips_and_updates Pro Tips

Matrix multiplication is NOT commutative: A×B ≠ B×A in general

For multiplication, the number of columns in A must equal the number of rows in B

A matrix has an inverse only if its determinant is non-zero

Transpose of a product: (AB)ᵀ = BᵀAᵀ (note the reversed order)

The determinant of a triangular matrix is the product of its diagonal elements

Row operations (swap, scale, add) don't change the solution set of a linear system

Essential Matrix Operations in Linear Algebra

Matrices are the workhorses of linear algebra, with applications spanning computer graphics, machine learning, structural engineering, quantum mechanics, and economics. A matrix is simply a rectangular array of numbers arranged in rows and columns, but the operations defined on them — addition, multiplication, inversion, determinants, and row reduction — form the mathematical backbone of modern computation. Matrix multiplication, unlike scalar multiplication, is not commutative (A times B does not generally equal B times A), and understanding this property is critical for correct application. The determinant of a square matrix tells you whether the matrix is invertible (determinant is nonzero) and represents the scaling factor the matrix applies to area or volume. For a 2x2 matrix, the determinant is ad minus bc; for larger matrices, cofactor expansion or row reduction is required. The inverse of a matrix, when it exists, solves systems of linear equations: if Ax = b, then x = A⁻¹b. Gaussian elimination (row reduction) is the most numerically stable method for solving these systems, reducing a matrix to row echelon form through systematic elimination of variables. These operations are computed billions of times per second in applications ranging from 3D rendering to training neural networks.

What is a Matrix Calculator?

A matrix calculator performs linear algebra operations on rectangular arrays of numbers. From simple addition to complex operations like finding inverses and row-reducing, our calculator handles all standard matrix operations with full step-by-step explanations.

Matrix Operations Overview

Addition/subtraction requires equal dimensions and works element-wise. Multiplication requires A's columns = B's rows. The determinant is a scalar that indicates invertibility. The inverse undoes multiplication (A × A⁻¹ = I). RREF uses Gauss-Jordan elimination to solve linear systems.

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