Matrix Calculator

Comprehensive matrix calculator for 2×2 and 3×3 matrices. Perform addition, subtraction, multiplication, find determinants, calculate inverses, and transpose with detailed solutions.

Matrix Calculator

Matrix A (2x2)
Matrix B (2x2)

Matrix Operations

Addition/Subtraction: Matrices must have the same dimensions
Multiplication: Number of columns in A must equal rows in B
Determinant: Only for square matrices (2×2 or 3×3)
Inverse: Only exists if determinant ≠ 0
Transpose: Rows become columns and vice versa

How to Use the Matrix Calculator

  1. Choose Operation: Select from addition, subtraction, multiplication, determinant, inverse, or transpose
  2. Select Matrix Size: Choose between 2×2 or 3×3 matrices depending on your problem
  3. Enter Matrix Elements: Fill in the matrix values in the grid input fields
  4. Two-Matrix Operations: For addition, subtraction, and multiplication, both Matrix A and B inputs will appear
  5. View Results: See the calculated result matrix along with step-by-step solution process
  6. Check Calculations: Review the detailed steps to understand the mathematical process

Examples and Use Cases

Matrix Operations

A + B = [1 2] + [5 6] = [6 8]
[3 4] [7 8] [10 12]
Matrix addition example
A × B = [1 2] × [5 6] = [19 22]
[3 4] [7 8] [43 50]
Matrix multiplication example

Advanced Operations

det([1 2]) = 1×4 - 2×3 = -2
([3 4])
2×2 determinant calculation
A^T = [1 3] (rows become columns)
[2 4]
Matrix transpose example

Understanding the Results

Operation Types

Addition & Subtraction

Element-wise operations. Matrices must have the same dimensions. Result[i][j] = A[i][j] ± B[i][j].

Multiplication

Row-by-column multiplication. Columns in A must equal rows in B. Result[i][j] = sum of A[i][k] × B[k][j].

Transpose

Flips matrix over its diagonal. Rows become columns and columns become rows.

Key Concepts

Determinant

Scalar value that indicates if a matrix has an inverse. Zero determinant means no inverse exists.

Matrix Inverse

A⁻¹ such that A × A⁻¹ = I (identity matrix). Only exists when determinant ≠ 0.

Cofactor Expansion

Method for calculating 3×3 determinants by expanding along a row or column.

Frequently Asked Questions

When can I multiply two matrices?

You can multiply matrix A × B only when the number of columns in A equals the number of rows in B. For example, a 2×3 matrix can be multiplied by a 3×2 matrix, resulting in a 2×2 matrix.

What does it mean when a matrix has no inverse?

A matrix has no inverse when its determinant equals zero. This means the matrix is "singular" or "non-invertible," and there's no matrix that can be multiplied with it to get the identity matrix.

How is the 3×3 determinant calculated?

The 3×3 determinant is calculated using cofactor expansion, typically along the first row. Each element is multiplied by its cofactor (the determinant of the 2×2 submatrix with alternating signs).

Can I use decimal numbers in matrices?

Yes! The calculator accepts decimal numbers and will perform all calculations accurately. Results are rounded to 4 decimal places for readability.

What is the difference between A×B and B×A?

Matrix multiplication is not commutative, meaning A×B ≠ B×A in general. The order matters because the calculation method (row × column) produces different results when the matrices are swapped.

How accurate are the calculations?

The calculator uses JavaScript's double-precision arithmetic for all operations. Results are mathematically accurate within the limits of floating-point precision and are displayed rounded to 4 decimal places.

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