Matrix Calculator
Add, subtract, multiply, and invert 2x2 and 3x3 matrices - with every step shown, from the determinant to the final inverse.
Calculate
Matrix A
Matrix B
Tap a cell and type. B is only needed for A+B, A−B and A×B.
Result
Step by step
- det = a·d − b·c
- det = 1·4 − 2·3
- det = -2
What Each Operation Means
Adding and subtracting matrices works cell by cell — the matrices must be the same size. Multiplying is richer: each cell of the answer is a whole row-times-column sum, which is why matrix multiplication doesn't commute (A×B usually differs from B×A).
The determinant is a single number that summarizes a matrix. If it's zero, the matrix squashes space flat and has no inverse — like trying to un-mix paint. The inverse A⁻¹ is the matrix that "undoes" A: A × A⁻¹ gives the identity matrix. The transpose simply flips rows and columns.
Where Matrices Show Up
Every 3D video game moves characters with matrix multiplication — rotation, scaling and translation are all matrices. Engineers solve big systems of equations with them, economists model input-output tables, and Google's original PageRank treated the whole web as one giant matrix. Learning the 2×2 and 3×3 cases by hand builds the intuition behind all of it.
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Save a number once — like your rent or hourly rate — then just type its name instead of the number, in any calculator here. You can also type formulas like 5000 * 1.1 right into any box.
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Frequently Asked Questions
How do you multiply two matrices?
Each cell of the answer is a row-times-column sum: C[i,j] = A[i,1]*B[1,j] + A[i,2]*B[2,j] + ... . That is why matrix multiplication does not commute - AxB usually differs from BxA.
What is a determinant?
A single number summarizing a matrix. For 2x2 [[a,b],[c,d]] it is ad - bc. If the determinant is zero, the matrix is singular: it squashes space flat and has no inverse.
How do you find the inverse of a 2x2 matrix?
Swap a and d, negate b and c, then divide everything by the determinant: 1/(ad-bc) * [[d,-b],[-c,a]]. If ad-bc = 0, no inverse exists.
What is a matrix transpose?
The transpose flips a matrix over its diagonal: rows become columns. T[i,j] = A[j,i].
Where are matrices actually used?
Everywhere in 3D graphics (rotations and positions are matrices), engineering (solving big equation systems), and economics (input-output models).