Matrix Calculator
Comprehensive matrix calculator for 2x2 and 3x3 matrices. Perform addition, subtraction, multiplication, find determinants, calculate inverses, and transpose with detailed solutions.
Matrix Calculator
Matrix Operations
How to Use the Matrix Calculator
- Choose Operation: Select from addition, subtraction, multiplication, determinant, inverse, or transpose
- Select Matrix Size: Choose between 2×2 or 3×3 matrices depending on your problem
- Enter Matrix Elements: Fill in the matrix values in the grid input fields
- Two-Matrix Operations: For addition, subtraction, and multiplication, both Matrix A and B inputs will appear
- View Results: See the calculated result matrix along with step-by-step solution process
- Check Calculations: Review the detailed steps to understand the mathematical process
Calculation Examples
Matrix Operations
Advanced Operations
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.
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.
Related Calculators
Scientific Calculator
Free scientific calculator with trigonometric, logarithmic, exponential functions. Includes sin, cos, tan, log, ln, powers, roots, constants π and e, memory functions.
Statistics Calculator
Free statistics calculator for data analysis. Calculate mean, median, mode, standard deviation, variance, range, quartiles and frequency distribution for any dataset.
Basic Calculator
Free basic calculator for addition, subtraction, multiplication, and division. Easy-to-use online calculator with history feature and responsive design.