Matrix Calculator
Perform the core matrix operations on 2×2 and 3×3 matrices: add, subtract and multiply, transpose, and find determinants and inverses.
Calculator
- 2×2 and 3×3
- Add, subtract, multiply
- Determinant, inverse, transpose
- Singular-matrix guard
- Inverse identity check
Result & calculation receipt
- Method
- 2×2 determinant by cofactor expansion
- Formula
det = ad − bc- Substitution
- det(A) = 10
- Steps
- 4·6 − 7·2 = 10
- Verification
- det ≠ 0 → the matrix is invertible.
- Exact result
- 10
- Decimal result
- 10
- Engine
- math-solvers v1.4.0
How this calculator works
You enter a matrix row by row (and a second one for binary operations). Addition and subtraction combine matching entries; multiplication takes row-by-column dot products; the transpose swaps rows and columns; the determinant uses cofactor expansion; and the inverse uses the adjugate divided by the determinant. A matrix with determinant zero is singular and has no inverse, which the calculator reports, and every inverse is verified by multiplying it back to the identity.
Worked example
The determinant of [[1, 2], [3, 4]] is 1·4 − 2·3 = −2. The inverse of [[4, 7], [2, 6]] is verified by multiplying it by the original to get the identity matrix. Multiplying [[1, 2], [3, 4]] by [[5, 6], [7, 8]] gives [[19, 22], [43, 50]], each entry a row-times-column dot product.
Key insight
The determinant is the single number that decides whether a matrix is invertible: non-zero means yes, zero means the matrix collapses space and cannot be undone. That is the same condition that decides whether a linear system has a unique solution.
Frequently asked questions
How do I multiply two matrices?
Each entry of the product is the dot product of a row of the first matrix with a column of the second. Multiplication is only defined when the first matrix’s column count equals the second’s row count; here both are square and the same size.
What is a determinant?
The determinant is a single number computed from a square matrix (ad − bc for 2×2, cofactor expansion for 3×3). It is zero exactly when the matrix is singular and has no inverse.
When does a matrix have no inverse?
When its determinant is zero. Such a matrix is called singular; the calculator detects this and returns an error instead of dividing by zero.
How is the inverse verified?
By multiplying the matrix by its computed inverse: the result should be the identity matrix (ones on the diagonal, zeros elsewhere). The calculator shows that product.
How do I enter a matrix?
List the entries row by row, separated by commas, spaces or new lines: a 2×2 needs four numbers, a 3×3 needs nine. A wrong count or a non-numeric entry returns an error.
Methodology & verification
Matrices are parsed row by row and must contain exactly size² numeric entries. Addition and subtraction are entrywise; multiplication uses row-by-column dot products; the transpose swaps indices; the determinant uses cofactor expansion; the inverse is the adjugate divided by the determinant. A zero determinant returns a singular-matrix error rather than dividing by zero. Every inverse is verified by multiplying it by the original matrix and confirming the product is the identity.
Engine: math-solvers v1.4.0 · Last reviewed: · Every result is computed client-side and never sent to a server. See the verification methodology and the scientific calculator validation.