Matrix Determinant Calculator
The determinant of a square matrix is a single number that summarizes many properties of the matrix. It determines whether the matrix is invertible (nonzero determinant), the scaling factor of the linear transformation the matrix represents, and the sign encodes orientation. For a 2x2 matrix [[a,b],[c,d]], the determinant is ad - bc. For a 3x3 matrix, it is computed by cofactor expansion along the first row: a(ei-fh) - b(di-fg) + c(dh-eg). Select the matrix size, enter the entries, and compute the determinant instantly.
Row 1
Row 2
Determinant formulas
2x2: det[[a,b],[c,d]] = ad - bc
3x3: det = a(ei-fh) - b(di-fg) + c(dh-eg)
where rows are [a,b,c], [d,e,f], [g,h,i]
The 3x3 formula is cofactor expansion along the first row. Each first-row entry multiplies the 2x2 minor formed by deleting that entry's row and column, with alternating signs +, -, +.
Properties of the determinant
- det(AB) = det(A) * det(B) for square matrices of the same size.
- det(A^T) = det(A): the determinant is unchanged by transposing.
- Swapping two rows changes the sign of the determinant.
- Adding a multiple of one row to another does not change the determinant.
- det(kA) = k^n * det(A) for an n x n matrix scaled by k.
Matrix determinant: frequently asked questions
What is the determinant of a matrix?
The determinant is a scalar value computed from the entries of a square matrix. For a 2x2 matrix [[a,b],[c,d]], det = ad - bc. The determinant encodes geometric information: |det| is the area (2x2) or volume (3x3) scaling factor of the linear transformation.
How is the 3x3 determinant calculated?
For a 3x3 matrix, expand along the first row: det = a(ei-fh) - b(di-fg) + c(dh-eg), where the matrix is [[a,b,c],[d,e,f],[g,h,i]]. Each term is a signed minor (cofactor) of the first-row entry.
What does it mean if the determinant is zero?
A zero determinant means the matrix is singular (not invertible). The rows (or columns) are linearly dependent, meaning at least one row can be expressed as a linear combination of the others. Geometrically, the transformation collapses the space to a lower dimension.
What is Cramer's rule?
Cramer's rule solves a system Ax = b by computing x_i = det(A_i) / det(A), where A_i is the matrix A with column i replaced by b. It requires det(A) not equal to zero. It is useful for small systems but computationally expensive for large ones.
What is the relationship between the determinant and the inverse?
A matrix A has an inverse if and only if det(A) is not zero. The inverse is A^(-1) = adj(A) / det(A), where adj(A) is the adjugate (transpose of the cofactor matrix). For a 2x2 matrix, A^(-1) = (1/det) * [[d,-b],[-c,a]].
Official sources
- NIST Digital Library of Mathematical Functions: dlmf.nist.gov.
- NIST, Mathematical topics: nist.gov/topics/mathematics.
Reviewed by the CalculatorHub team, edited by James Graham, 15 June 2026. See our methodology.