Matrix Calculator – Add, Subtract, Multiply, Transpose, Determinant, and Inverse

  • Adam
  • November 4, 2025

Free online matrix calculator for matrix operations including addition, subtraction, multiplication, transpose, determinant, and inversion. Perform linear algebra calculations with step-by-step solutions.

Matrix Calculator

Advanced Tool for Matrix Operations and Linear Algebra

Quick Navigation

Matrix Addition

Add two matrices of equal dimensions

Rows: Columns:

Matrix Subtraction

Subtract two matrices of equal dimensions

Rows: Columns:

Matrix Multiplication

Multiply two matrices (columns of A = rows of B)

Rows: Columns:
Rows: Columns:

Matrix Transpose

Swap rows and columns

Rows: Columns:

Matrix Determinant

Calculate determinant of square matrix

Matrix Inverse

Calculate inverse of square matrix

What are Matrices?

A matrix is a rectangular array of numbers arranged in rows and columns. For example, a 2×3 matrix has 2 rows and 3 columns. Matrices are fundamental tools in mathematics, physics, computer science, and engineering for representing and manipulating data, solving systems of equations, and performing transformations.

Matrices are denoted using capital letters like A, B, C and individual elements are referenced by their position. For instance, element a₂₃ refers to the element in the second row and third column. Special types of matrices include the identity matrix (1s on diagonal), zero matrix (all 0s), and square matrices (same number of rows and columns).

Understanding matrix operations is essential for linear algebra, computer graphics, image processing, data science, and physics simulations. This calculator helps you perform various matrix operations with complete step-by-step explanations, making it easy to learn and verify your calculations.

Key Concept: Matrices follow specific rules for operations. Not all matrix operations are possible between all matrix pairs—dimensions must satisfy certain conditions.

Key Features & Capabilities

This comprehensive matrix calculator provides multiple matrix operations with detailed analysis:

➕ Matrix Addition Add two matrices of equal dimensions
➖ Matrix Subtraction Subtract two matrices of equal dimensions
✕ Matrix Multiplication Multiply matrices with compatible dimensions
🔄 Matrix Transpose Swap rows and columns of a matrix
🔢 Determinant Calculate determinant of square matrices
↩️ Matrix Inverse Find inverse of square matrices
📋 Flexible Dimensions Work with matrices up to 5×5 size
📋 Step-by-Step Solutions Detailed breakdown of each operation
✓ Validation Checks Verifies dimension compatibility
📋 Copy to Clipboard One-click copy functionality
🎓 Educational Content Comprehensive guides and examples
📱 Fully Responsive Works seamlessly on all devices

How to Use This Calculator

Step-by-Step Guide

  1. Choose Operation: Select the matrix operation you need: Addition, Subtraction, Multiplication, Transpose, Determinant, or Inverse.
  2. Set Dimensions: Specify the dimensions (rows and columns) for your matrices. The calculator will create input fields automatically.
  3. Enter Values: Input the numerical values for each matrix element in the grid. Use Tab or arrow keys to move between cells.
  4. Check Compatibility: Ensure dimensions are compatible for your operation (e.g., for multiplication, columns of A must equal rows of B).
  5. Click Calculate: Press the Calculate button to perform the matrix operation.
  6. Review Results: See the result matrix displayed in organized format.
  7. Study Steps: Understand how the operation was performed with detailed breakdown.
  8. Copy or Clear: Use Copy for results or Clear to start a new calculation.

Tips for Accurate Use

  • Dimension Compatibility: For addition/subtraction, matrices must have identical dimensions.
  • Multiplication Rule: Columns of first matrix must equal rows of second matrix.
  • Square Matrices: Determinant and inverse require square matrices (n×n).
  • Decimal Input: All calculators support decimal and negative numbers.
  • Identity Check: For inverse, verify determinant is non-zero first.

Complete Formulas Guide

Matrix Addition and Subtraction

Element-Wise Operations
(A + B)ᵢⱼ = Aᵢⱼ + Bᵢⱼ
(A - B)ᵢⱼ = Aᵢⱼ - Bᵢⱼ

Both matrices must have identical dimensions

Example: If A is 2×3 and B is 2×3, then A+B is 2×3

Matrix Multiplication

Dot Product of Rows and Columns
(AB)ᵢⱼ = Σ Aᵢₖ × Bₖⱼ (k=1 to n)

Columns of A must equal rows of B
Result: (m×n) × (n×p) = (m×p) matrix

Example: (2×3) × (3×2) = (2×2)

Matrix Transpose

Swap Rows and Columns
Aᵀᵢⱼ = Aⱼᵢ

If A is m×n, then Aᵀ is n×m
Row i becomes column i

Example: 2×3 matrix transposed becomes 3×2

Determinant (2×2)

2×2 Matrix Determinant
det(A) = ad - bc

For matrix [[a, b], [c, d]]

Example: [[2, 3], [4, 5]]
det = 2(5) - 3(4) = 10 - 12 = -2

Matrix Inverse (2×2)

2×2 Inverse Formula
A⁻¹ = (1/det(A)) × [[d, -b], [-c, a]]

Only exists if det(A) ≠ 0
A × A⁻¹ = I (identity matrix)

Matrix Operations Explained

Addition and Subtraction

Matrix addition and subtraction are performed element-wise, meaning each element in one matrix is added or subtracted from the corresponding element in the other matrix. Both matrices must have identical dimensions (same number of rows and columns). These are straightforward operations similar to adding or subtracting regular numbers.

Multiplication

Matrix multiplication is more complex than element-wise operations. The element in row i and column j of the result is calculated by taking the dot product of row i from the first matrix and column j from the second matrix. The number of columns in the first matrix must equal the number of rows in the second matrix for multiplication to be possible.

Transpose

The transpose of a matrix is obtained by interchanging its rows and columns. If element aᵢⱼ is in row i and column j, then in the transpose it will be in row j and column i. The transpose is denoted by Aᵀ or A'. An m×n matrix becomes an n×m matrix after transposition.

Determinant

The determinant is a scalar value calculated from a square matrix. It provides information about the matrix's properties—whether it's invertible, whether it transforms space, and the volume scaling factor. A determinant of zero means the matrix is singular and cannot be inverted.

Inverse

The inverse of a square matrix A is another matrix A⁻¹ such that A × A⁻¹ = I (the identity matrix). Only square matrices with non-zero determinant can have an inverse. The inverse is used to solve matrix equations similar to how division works with numbers.

Worked Examples

Example 1: Matrix Addition

Problem: Add matrices A and B

Solution:
A = [[1, 2], [3, 4]]
B = [[2, 0], [1, 2]]

A + B = [[1+2, 2+0], [3+1, 4+2]]
Result = [[3, 2], [4, 6]]

Example 2: Matrix Multiplication

Problem: Multiply 2×2 matrices

Solution:
A = [[1, 2], [3, 4]]
B = [[5, 6], [7, 8]]

(AB)₁₁ = 1×5 + 2×7 = 5 + 14 = 19
(AB)₁₂ = 1×6 + 2×8 = 6 + 16 = 22
(AB)₂₁ = 3×5 + 4×7 = 15 + 28 = 43
(AB)₂₂ = 3×6 + 4×8 = 18 + 32 = 50

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

Example 3: Matrix Transpose

Problem: Find transpose of 2×3 matrix

Solution:
A = [[1, 2, 3], [4, 5, 6]]

Aᵀ = [[1, 4], [2, 5], [3, 6]]

Rows become columns: 2×3 → 3×2

Example 4: Calculate Determinant

Problem: Find determinant of 2×2 matrix

Solution:
A = [[2, 3], [4, 5]]

det(A) = (2)(5) - (3)(4)
= 10 - 12
= -2

Example 5: Matrix Inverse

Problem: Find inverse of 2×2 matrix

Solution:
A = [[4, 7], [2, 6]]

det(A) = 4(6) - 7(2) = 24 - 14 = 10

A⁻¹ = (1/10) × [[6, -7], [-2, 4]]
= [[0.6, -0.7], [-0.2, 0.4]]

Frequently Asked Questions

❓ Can I multiply matrices of different dimensions?
Yes, but only if compatible: columns of first = rows of second. A 2×3 matrix can multiply with a 3×4 matrix to give a 2×4 result. The rule is (m×n) × (n×p) = (m×p).
❓ Is matrix multiplication commutative?
No. AB ≠ BA in general. Matrix multiplication is not commutative, meaning order matters. This is one key difference from regular number multiplication.
❓ What is an identity matrix?
The identity matrix I has 1s on the diagonal and 0s elsewhere. For any matrix A: AI = IA = A. It's the matrix equivalent of the number 1 in multiplication.
❓ When can't a matrix be inverted?
A matrix cannot be inverted if: (1) it's not square, (2) its determinant is zero (singular matrix), or (3) its rows/columns are linearly dependent. Check the determinant first.
❓ What does matrix transposition do?
Transposition swaps rows and columns. If element is at position (i,j), it moves to (j,i). It's useful in many applications including solving systems of equations and computer graphics.
❓ Can I add matrices of different sizes?
No. Matrices must have identical dimensions to be added or subtracted. You cannot add a 2×3 matrix to a 3×2 matrix—they must have the same number of rows and columns.
❓ What is a singular matrix?
A singular matrix is a square matrix with determinant equal to zero. Singular matrices cannot be inverted and don't have full rank. They represent transformations that collapse space into lower dimensions.
❓ When would I use matrices?
Matrices are used in: solving systems of linear equations, computer graphics transformations, image processing, neural networks, data analysis, physics simulations, and countless engineering applications.
❓ What's the difference between matrix transpose and inverse?
Transpose swaps rows and columns, while inverse is a matrix that satisfies A × A⁻¹ = I. They're completely different operations. Some matrices (orthogonal matrices) have Aᵀ = A⁻¹.

Start Calculating Matrices

Whether you're studying linear algebra, solving systems of equations, working with graphics transformations, or learning data science, this comprehensive matrix calculator provides instant solutions with complete analysis. Fast, accurate, and completely free.

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