Matrix Calculator
Calculate matrix multiplication, addition, subtraction, scalar multiplication, transpose, determinant, inverse, and RREF with dimension checks.
Last Updated: May 2026
A x B
3 x 3
A Dimensions
3 x 3
B Dimensions
3 x 3
Trace
71
Result matrix
10 3 14 28 9 32 47 15 52
| Detail | Value |
|---|---|
| Formula | c_ij = sum(a_ik x b_kj) |
| Rows in result | 3 |
| Columns in result | 3 |
| Minimum entry | 3 |
| Maximum entry | 52 |
| Step | Explanation |
|---|---|
| 1 | A is 3 x 3 and B is 3 x 3, so the product is 3 x 3. |
Addition requires matching dimensions. Multiplication requires columns of A to match rows of B.
Determinants and inverses only apply to square matrices.
RREF uses row operations and is useful for solving systems and checking rank.
Numerical Linear Algebra Notice
This calculator uses decimal arithmetic and rounding for display. Very ill-conditioned or nearly singular matrices may need exact symbolic tools or specialized numerical software for final coursework, engineering, or research use.
Reviewed For Methodology, Labels, And Sources
Every CalculatorWallah calculator is published with visible update labeling, linked source references, and founder-led review of formula clarity on trust-sensitive topics. Use results as planning support, then verify institution-, policy-, or jurisdiction-specific rules where they apply.
Reviewed By
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Page updated May 2026. Trust-critical pages are reviewed when official rates or rules change. Evergreen calculator guides are checked on a recurring quarterly or annual cycle depending on topic volatility.
How to Use the Matrix Calculator
Choose the matrix operation, then paste Matrix A and Matrix B when needed. Put each row on its own line and separate entries with spaces, commas, semicolons, or tabs.
Use determinant and inverse modes for square matrices. Use RREF mode for row reduction or augmented systems where constants appear in the final column.
Step 1: Choose an operation
Select addition, subtraction, multiplication, scalar multiplication, transpose, determinant, inverse, or RREF.
Step 2: Enter matrix rows
Type each row on a new line and separate entries consistently.
Step 3: Check dimensions
Use the result cards and error messages to confirm the operation is valid.
Step 4: Review the output
Read the result matrix, scalar output, formula, and row-operation notes.
How This Matrix Calculator Works
Element-wise operations combine matching matrix entries. Matrix multiplication uses row-by-column dot products, so the inner dimensions must match.
Determinant, inverse, and RREF calculations use Gaussian-elimination style row operations. Inverse mode augments Matrix A with the identity matrix and row-reduces to isolate the inverse.
Results are rounded for readability, and very small values near zero are displayed as zero to reduce numerical noise.
Matrix Operation Guide
Matrix Operation Rules
| Operation | Requirement | Meaning |
|---|---|---|
| Addition / subtraction | same dimensions | Combine matching entries element by element. |
| Multiplication | columns of A = rows of B | Each result entry is a row-by-column dot product. |
| Scalar multiply | any matrix | Multiply every entry by the same scalar. |
| Transpose | any matrix | Rows become columns and columns become rows. |
| Determinant | square matrix only | Returns a scalar used for invertibility and scaling checks. |
| Inverse | square, nonsingular matrix | A times A inverse equals the identity matrix. |
| RREF | any rectangular matrix | Uses row operations to simplify a matrix or augmented system. |
Input Format
| Input part | Rule | Example |
|---|---|---|
| Rows | Put each matrix row on its own line. | Example: first row on line 1, second row on line 2. |
| Columns | Separate entries with spaces, commas, semicolons, or tabs. | Example: 1 2 3 or 1, 2, 3. |
| Decimals | Decimal entries are accepted. | Results are rounded for display. |
| Augmented systems | Use RREF mode with constants as the final column. | Useful for linear systems. |
Most matrix mistakes are dimension mistakes. Check rows and columns before doing multiplication, and remember that A x B usually differs from B x A.
For equation-level algebra, use the math equation solver. For numeric checks around entries or determinants, use the scientific calculator.
Keep the research moving with Scientific Calculator, Math Equation Solver, Statistics Calculator, and Probability Calculator.
Frequently Asked Questions
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Use Fraction CalculatorSources & References
- 1.OpenStax Algebra and Trigonometry - Gaussian Elimination(Accessed May 2026)
- 2.NIST Engineering Statistics Handbook - Determinant and Eigenstructure(Accessed May 2026)
- 3.NIST JAMA Matrix Class Reference(Accessed May 2026)