Math
Matrix Calculator
Add, subtract, multiply, and find the determinant, inverse, or transpose of 2×2 and 3×3 matrices.
Matrix A
Matrix B
Result
Formulas
Addition
(A + B)ᵢⱼ = Aᵢⱼ + Bᵢⱼ
Add corresponding elements
Subtraction
(A − B)ᵢⱼ = Aᵢⱼ − Bᵢⱼ
Subtract corresponding elements
Multiplication
(A×B)ᵢⱼ = Σₖ Aᵢₖ·Bₖⱼ
Dot product of rows and columns
Det. 2×2
det = ad − bc
For [[a,b],[c,d]]
Det. 3×3
a(ei−fh) − b(di−fg) + c(dh−eg)
Cofactor expansion along row 1
Inverse
A⁻¹ = (1/det) · adj(A)
Undefined when det = 0
Transpose
Aᵀᵢⱼ = Aⱼᵢ
Swap rows and columns
Why the Determinant Matters
Non-zero determinant
The matrix is invertible ("non-singular"). Its rows and columns are linearly independent, and any linear system Ax = b has exactly one solution.
Zero determinant
The matrix is singular — no inverse exists. The system Ax = b has either no solution or infinitely many.
The inverse is calculated with the adjugate method: transpose the cofactor matrix, then divide every entry by the determinant.
Tips
Multiplication order matters: A×B is generally not equal to B×A, even for square matrices of the same size.
Transpose is free: it never fails, unlike inverse, which requires a non-zero determinant.
Check your work: multiplying A by its inverse A⁻¹ should return the identity matrix (1s on the diagonal, 0s elsewhere).
FAQ
Frequently asked questions.
What operations can this matrix calculator perform?
This calculator performs the core matrix operations you need for algebra and linear systems: addition, subtraction, and multiplication of two matrices, plus determinant, inverse, and transpose of a single matrix. It supports both 2×2 and 3×3 square matrices, so you can quickly check homework, verify hand calculations, or explore how each operation changes a matrix without any software installation.
How does matrix multiplication work?
Matrix multiplication combines rows of the first matrix with columns of the second using the dot product: each entry in the result is the sum of the products of corresponding row and column elements. Unlike addition, matrix multiplication is not commutative, meaning A×B usually does not equal B×A. In general, two matrices can only be multiplied when the number of columns in the first equals the number of rows in the second — in this calculator both matrices are square and the same size, so that requirement is always satisfied.
What does the determinant of a matrix tell you?
The determinant is a single number that summarizes key properties of a square matrix. If the determinant is zero, the matrix is called singular, meaning it has no inverse and the corresponding system of linear equations does not have a unique solution. A non-zero determinant confirms the matrix is invertible and its rows (or columns) are linearly independent. Geometrically, the absolute value of the determinant also represents how much the matrix scales area (2×2) or volume (3×3).
How is the matrix inverse used?
The inverse of a matrix A, written A⁻¹, is the matrix that returns the identity matrix when multiplied by A. Inverses are most commonly used to solve systems of linear equations: if Ax = b, then x = A⁻¹b gives the solution directly. This calculator computes the inverse using the adjugate (cofactor) method for both 2×2 and 3×3 matrices, and will tell you when a matrix is singular and has no inverse.
What is the difference between transpose and inverse?
The transpose of a matrix simply swaps its rows and columns — no arithmetic is involved, and every matrix (square or not) has a transpose. The inverse, by contrast, is a matrix that undoes the original matrix's transformation through multiplication, requires the matrix to be square with a non-zero determinant, and involves solving for cofactors and dividing by the determinant. Transpose is a structural rearrangement; inverse is a computed algebraic counterpart that only exists for non-singular matrices.
Can I use this calculator for both 2×2 and 3×3 matrices?
Yes. Use the size selector to switch between 2×2 and 3×3 matrices, and the input grid updates instantly to match. All six operations — addition, subtraction, multiplication, determinant, inverse, and transpose — are supported at both sizes, with the determinant and inverse calculations using the appropriate formulas for each dimension.
Last updated: August 17, 2026