Matrix Calculator — Multiply, Invert & Reduce Matrices Step by Step
Enter two 2×2 or 3×3 matrices below to add, subtract, multiply, transpose, find a determinant, invert, or row-reduce them. Every result shows the arithmetic, not just the final grid of numbers.
Matrix size
Matrix A
Matrix B
Combine A and B
Single-matrix operations — apply to:
Fill in Matrix A and Matrix B, then pick an operation. Addition, subtraction, and multiplication combine both matrices; determinant, inverse, transpose, and row reduction work on whichever single matrix you select from the dropdown.
This covers 2×2 and 3×3 matrices, which is what most algebra, precalculus, and intro linear algebra courses actually use. For eigenvalues, decompositions, and other advanced operations, see the guidance section further down the page.
Reviewed by Amanda Reeds, Content Researcher · AceCalculator
What a Matrix Calculator Actually Does
A matrix calculator performs arithmetic on grids of numbers, called matrices, using rules different from ordinary algebra: addition and subtraction work position by position, multiplication combines rows and columns, and only square matrices with a nonzero determinant can be inverted. Students, engineers, and data analysts use it to solve systems of equations, transform coordinates, and check hand calculations.
A single-variable equation like 2x + 3 = 11 needs algebra. A system with three variables and three equations is usually faster to solve as a matrix, which is exactly what row reduction below is built for.
Matrix Operations This Page Helps With
Add & Subtract Matrices
Combine two matrices of the same size position by position. The widget above handles this instantly for 2×2 and 3×3 grids.
- Requires matching dimensions
- Also covers vector addition and subtraction
Multiply Matrices
A×B and B×A usually give different results, so the widget calculates both directions separately rather than assuming they match.
- Row-by-column multiplication
- Order matters: not commutative
Find a Determinant
A single number that tells you whether a matrix can be inverted (nonzero) and shows up in area, volume, and system-of-equations calculations.
- 2×2: cross-multiply and subtract
- 3×3: cofactor expansion
Find the Inverse
The inverse "undoes" a matrix the way dividing by a number undoes multiplying by it. Only square matrices with a nonzero determinant have one.
- 2×2 and 3×3 supported
- Flags singular (non-invertible) matrices
Transpose a Matrix
Flip a matrix over its diagonal so rows become columns. Useful for checking symmetry and as a step inside the inverse formula itself.
- Works for any rectangular matrix
- Instant, no determinant required
Solve Systems by Row Reduction
Row-reduce a matrix to reduced row echelon form (RREF) using Gauss-Jordan elimination, the same method behind solving multi-variable systems of equations.
- Gaussian elimination with pivoting
- Works for augmented (system) matrices too
Which Matrix Operation Do You Need?
Solving a system with 2 or 3 variables?
Build the coefficients into a matrix, add the answer column as an augmented matrix, and run row reduction (RREF) above. The final column shows each variable's value directly once it reaches identity form on the left.
Working with vectors instead of a full grid?
A vector is just a matrix with one column (or one row). Enter it as an n×1 shape in the calculator above and use addition, subtraction, or multiplication the same way.
Need to know if a matrix can be inverted?
Check the determinant first. If it's zero, the matrix is singular and has no inverse — the calculator above will tell you this directly instead of returning a broken result.
Watch Out: Not Every Matrix Operation Is Built Yet
Eigenvalues, eigenvectors, LU and QR decomposition, Jacobian and Hessian matrices, and Laplace transforms are all real linear algebra tools, but this calculator doesn't compute them yet. Check our other calculators hub for what's currently available, or search for a dedicated eigenvalue calculator in the meantime.
Matrix Formula Reference
Pro Tip
If the calculator says a matrix is singular, double-check for a repeated or all-zero row before assuming you made an arithmetic mistake. A row that's a multiple of another row always produces a determinant of zero.
About This Calculator
AceCalculator built the widget above using the same cofactor-expansion and Gauss-Jordan methods taught in a standard linear algebra course, then checked every function against known matrices before publishing — including confirming that a matrix multiplied by its own inverse returns the identity matrix.
It's scoped to 2×2 and 3×3 matrices, which cover the large majority of algebra and intro linear algebra coursework. For single-variable equations, our algebra calculator (linked above) is the better fit. See our About page for more on how AceCalculator's tools are built and checked.
Who This Page Is For
- Algebra 2 and precalculus students learning matrix basics
- Intro linear algebra students checking homework
- Engineering and computer science students working with transformations
- Anyone solving a multi-variable system of equations
- Tutors who want a quick step-by-step reference to point to
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Frequently Asked Questions About Matrix Calculators
How do you find the inverse of a 3x3 matrix?
Calculate the determinant first; if it's zero, no inverse exists. Otherwise, build the cofactor matrix, transpose it to get the adjugate, then divide every entry by the determinant. The calculator above does this automatically and shows the result.
What is row echelon form used for?
Row echelon form simplifies a matrix so each row's leading number sits further right than the row above it, which makes solving a system of equations, finding rank, or spotting dependent rows much faster than working with the original matrix.
What's the difference between a determinant and an inverse?
A determinant is a single number describing a matrix, used mainly to check whether it can be inverted. An inverse is an entirely new matrix that, when multiplied by the original, produces the identity matrix.
Why does a matrix need to be square to have an inverse?
The inverse has to "undo" a transformation back to its original dimensions, which is only mathematically possible when the number of rows equals the number of columns. Non-square matrices use different tools, like the pseudoinverse, instead.
Does this calculator find eigenvalues or eigenvectors?
Not yet. Eigenvalues and eigenvectors require solving a matrix's characteristic polynomial, which is a separate calculation from the addition, multiplication, determinant, inverse, and row-reduction operations this tool currently supports.
Is this the same as Symbolab or Wolfram's matrix calculator?
No, this is AceCalculator's own tool, built independently. Symbolab and Wolfram Alpha are separate platforms with broader symbolic math engines; all cover core matrix operations like determinants and inverses, so comparing results is a reasonable way to double-check work.
Can I solve a system of equations using this matrix calculator?
Yes. Enter the system's coefficients as a matrix with the constant terms as an extra column, then use the row reduction (RREF) button. Once the left side reaches identity form, the right column shows each variable's solved value.
Disclaimer
This tool is for informational and educational purposes only. Symbolab, Wolfram Alpha, Desmos, Reshish, Texas Instruments, and other named products are trademarks of their respective owners; AceCalculator is not affiliated with or endorsed by them. Always double-check important calculations, especially for graded work.